\section{Introduction}


In the study of statistical inference for parametric models, {\it
asymptotic efficiency} {\color{black}}% {\color[rgb]{0,0,0.8} plays a key role} when we consider {\color{black} the	} asymptotic
optimality of estimators.
This notion {\color{black} was} first studied for models {\color{black} that} satisfy {\it local asymptotic normality} (LAN);
H\'ajek \cite{haj70} showed the convolution theorem, and H\'ajek \cite{haj72} showed the minimax theorem under the LAN property.
Both theorems give different concepts of asymptotic efficiency.
For statistical models with {\color{black} the extended notion of} {\it local asymptotic mixed normality} (LAMN),
Jeganathan \cite{jeg82,jeg83} showed the convolution theorem and the minimax theorem.

Gobet \cite{gob01} showed the LAMN property for discretely observed diffusion processes on a fixed interval.
In {\color{black} that} model, the maximum-likelihood-type estimator proposed {\color{black} by} Genon-Catalot and Jacod~\cite{gen-jac93} is asymptotically efficient.
{\color{black} For} further results related to diffusion processes on a fixed interval, 
see Gloter and Jacod~\cite{glo-jac01a} (LAN for noisy observations of diffusion processes with deterministic diffusion coefficients),
Gloter and Gobet~\cite{glo-gob08} (LAMN for integrated diffusion processes),
Ogihara~\cite{ogi15} (LAMN for nonsynchronously observed diffusion processes),
and Ogihara~\cite{ogi18} (LAN for noisy, nonsynchronous observations of diffusion processes with deterministic diffusion coefficients).

In the model of discretely observed diffusion processes {\color{black} by} Gobet~\cite{gob01}, 
he initiated a scheme based on Malliavin calculus techniques to show the LAMN property.
{\color{black} He introduced Malliavin calculus techniques to control {\color{black} the} asymptotics of log-likelihood ratios,
and his scheme is {\color{black} effective} for diffusion processes when the diffusion coefficient matrix is nondegenerate.}
In his scheme, it is crucial that the transition density functions of diffusion processes are estimated from below and above
by Gaussian density functions.
{\color{black} Such estimates are {\color{black} known as} Aronson's estimate.}
Gloter and Gobet \cite{glo-gob08} {\color{black}}% {\color[rgb]{0,0,0.8} gave} {\color{black} Aronson's estimate}, and consequently showed
the LAMN property for the one-dimensional integrated diffusion processes by using Gobet's scheme.
However, {\color{black} the proof} of {\color{black} Aronson's estimate} (Theorem 4) {\color{black} crucially} depends on 
the fact that the latent process is one-dimensional.

For {\color{black} a diffusion model with the degenerate diffusion coefficient and} a multi-dimensional integrated diffusion model, it seems difficult
to obtain {\color{black} Aronson's estimate} {\color{black} in general}, and therefore we cannot apply Gobet's scheme.
On the other hand, Theorem 1 in Jeganathan~\cite{jeg82} introduced a scheme by using {\color{black} the} so-called $L^2$ regularity condition to show the LAMN property.
{\color{black} An advantage} of this scheme is that we do not need estimates for
the transition density functions.
{\color{black} However, the} results in \cite{jeg82} 
are not directly applicable to {\color{black} high-frequency} observations
that require a framework of triangular arrays.
Further, for integrated diffusions, 
following the idea in \cite{glo-gob08}, 
we need to consider a triangular array of expanding data blocks.

This paper studies {\color{black} four} topics. First, we extend Theorem~1 in~\cite{jeg82} so that it can be applied to statistical {\color{black} models} with triangular array observations
appearing in the above diffusion models with high-frequency observations.
Second, we show that the new scheme based on the $L^2$ regularity condition can be applied under several conditions described via notions of Malliavin calculus.
The new scheme is {\color{black} highly} compatible with Gobet's scheme. {\color{black} Indeed}, the $L^2$ regularity condition is satisfied when observations are smooth in the Malliavin sense,
and the inverse of Malliavin matrix and its derivatives have moments (see (B1), (B2), and Theorem~\ref{Malliavin-LAMN-thm2}).
Moreover, if observations have {\color{black} a Euler--Maruyama} approximation, {\color{black} then the} sufficient conditions for the LAMN property is simplified (Theorem~\ref{Malliavin-LAMN-thm}).
Third, by using these schemes, we show the LAMN property for {\color{black} diffusion processes with the degenerate diffusion coefficient (degenerate diffusion)
in which it is difficult to obtain Aronson-type {\color{black} estimates} in general.
{\color{black} Finally}, we deal with the LAMN property for partial observations of degenerate diffusion processes.}


{\color{black} Our} new schemes can be applied to general statistical models without transition density estimates. 
In particular, they can be applied even when the transition density
function has zero points.
{\color{black} The $L^2$ regularity condition is related to differences {\color{black} in the roots} of transition density functions,
and it is not easily applied when the transition density functions have zero points (see (\ref{L2-regu-equiv})).
However, we will see in Section~\ref{LAMN-Malliavin-subsection} that if {\color{black} the} observations are smooth in {\color{black} the} Malliavin's sense 
and the Malliavin matrix is nondegenerate, {\color{black} then} we can apply the $L^2$ regularity condition even when the transition density has zero points.
{\color{black} Consequently}, this scheme enables us to study the LAMN property for several statistical models in Wiener space.
First, this scheme {\color{black} allows a} simplified proof of the results in Gobet~\cite{gob01}.
Moreover, this scheme yields two interesting results.
The first one is an extension of the results in Gobet~\cite{gob01} to a wider class including degenerate diffusion processes;
we emphasis that this is achieved {\color{black} because} we do not rely on Aronson-type estimates.
The second one is an extension of the LAMN property for one-dimensional integrated diffusion processes 
in Gloter and Gobet~\cite{glo-gob08} to the multi-dimensional case.
We deal with the integrated diffusion process model in {\color{black} the} general framework of partial observations for degenerate diffusion processes.
We {\color{black} find} that efficient asymptotic variance {\color{black} is the same for an} integrated diffusion process model and for {\color{black} a} diffusion process model,
and {\color{black} that} they are exactly twice as large as the statistical model of both observations (see Remark~\ref{integrated-rem}). 
{\color{black} Because} our scheme does not require transition density estimates, 
{\color{black} we expect these ideas to be useful also} for jump-diffusion process models or L\'evy driven {\color{black} stochastic differential equation} models.
However, we left this {\color{black} for} future work.}

{\color{black}}% {\color[rgb]{0,0,0.8} Our study of integrated diffusion models is motivated by 
{\color{black} experimental observations of single molecules}
(see e.g. Li et al.~\cite{liEtAl10}), {\color{black} behind which are Langevin-type} molecular dynamics
\begin{equation*}
\ddot{Y} = b(\dot{Y},Y) + {\color{black} a(\dot{Y})}\dot{W}.
\end{equation*}
Here $Y$ represents the position of a molecule (or a particle) and $\dot{W}$ is {\color{black} white}
noise.
When $a = 0$ this reduces to the {\color{black} Newtonian} equation of classical dynamics.
The system can be written as an integrated diffusion
\begin{equation}\label{Langevin-eq}
\begin{split}
 & dY_t = X_t dt, \\
& dX_t = b(X_t,Y_t)dt + {\color{black} a(X_t)}dW_t.
 \end{split}
\end{equation}
Our LAMN property enables us to discuss optimality in estimating the
coefficient $a$ 
based on high-frequency observations of the position $Y$.

}

The rest of this paper is organized as follows. In Section~\ref{main-results-section}, we introduce {\color{black} our} main results, {\color{black} namely},
the extended scheme using the $L^2$ regularity condition, the scheme via Malliavin calculus techniques, 
and the LAMN property of {\color{black} degenerate} diffusion processes.
Section~\ref{Mcal-section} contains details of Malliavin calculus techniques. We combine the extended scheme of {\color{black} the} $L^2$ regularity condition
with {\color{black} the} approaches of Gobet~\cite{gob01} and Gloter and Gobet~\cite{glo-gob08}.

\section{Main results}\label{main-results-section}

\subsection{The LAMN {\color{black} property} via the $L^2$ regularity condition}\label{L2regu-subsection}

In this subsection, we extend Theorem~1 in Jeganathan~\cite{jeg82} to statistical models of triangular array observations
so that it can be applied to high-frequency observations of stochastic processes.

Let $\{P_{\theta,n}\}_{\theta\in\Theta}$ be a family of probability
 measures defined on $({\color{black} \mathfrak{R}_n},\mathcal{F}_n)$,
where $\Theta$ is an open subset of $\mathbb{R}^d$. 
We often regard a $p$-dimensional vector $v$ as a $p\times 1$ matrix.
{\color{black} $I_k$ denotes the unit matrix of size $k$ for $k\in\mathbb{N}$.}

{\color{black}
\begin{description}
\item[{\color{black} Condition} (L).] The following two conditions are satisfied for $\{P_{\theta,n}\}_{\theta\in\Theta}$.
\begin{enumerate}
\item 
There exists a sequence $\{V_n(\theta_0)\}$ of $\mathcal{F}_n$-measurable $d$-dimensional vectors and a sequence $\{T_n(\theta_0)\}$
of $\mathcal{F}_n$-measurable $d\times d$ symmetric matrices such that 
\begin{equation}\label{Tn-nd}
P_{\theta_0,n}[T_n(\theta_0) {\rm \ is \  nonnegative \ definite}]=1
\end{equation}
for any $n\in\mathbb{N}$, and
\begin{equation}\label{log-like-est2}
\log \frac{dP_{\theta_0+r_nh,n}}{dP_{\theta_0,n}}-h^{\top}V_n(\theta_0)+\frac{1}{2}h^{\top}T_n(\theta_0)h\to 0
\end{equation}
in $P_{\theta_0,n}$-probability for any $h\in\mathbb{R}^d$, where
$\{r_n\}$ is a sequence of positive definite matrices
and $\top$ denotes the transpose operator for matrices.
\item {\color{black} There} exists an almost surely nonnegative definite random matrix $T(\theta_0)$ such that
\begin{equation*}
\mathcal{L}(V_n(\theta_0),T_n(\theta_0)|P_{\theta_0,n})\to \mathcal{L}(T^{1/2}(\theta_0)W,T(\theta_0)),
\end{equation*}
where $W$ is a $d$-dimensional standard normal random variable independent of $T(\theta_0)$.
\end{enumerate}
\end{description}
}

The following definition of the LAMN property is Definition~1 in~\cite{jeg82}.
\begin{definition}\label{LAMN-def}
The sequence of the families $\{P_{\theta,n}\}_{\theta\in\Theta} \ (n\in\mathbb{N})$
satisfies the LAMN condition at $\theta=\theta_0\in\Theta$ if {\color{black} Condition (L) is satisfied, 
$P_{\theta_0,n}[T_n(\theta_0) {\rm \ is \  positive \ definite}]=1$
for any $n\in\mathbb{N}$,
and $T(\theta_0)$ is positive definite almost surely.}
\end{definition}

{\color{black} For proving} the LAMN property for diffusion processes using a localization technique {\color{black} such as} Lemma 4.1 in~\cite{gob01},
Condition~(L) is useful {\color{black} because} (L) for the localized model often implies (L) for the original model.
See the proofs of Theorems~\ref{degenerate-LAMN-thm} and~\ref{partial-LAMN-thm} for the details. 
\begin{remark}\label{Gamma-pd-rem}
When Condition~(L) is satisfied and $T(\theta_0)$ is positive definite almost surely, by setting
\begin{equation}
{\color{black} \tilde{T}_n(\theta_0)=T_n(\theta_0)1_{\{T_n(\theta_0) \ {\rm is \ p.d.}\}}+I_d1_{\{T_n(\theta_0) \ {\rm is \ not \ p.d.}\}},}
\end{equation}
the LAMN property holds with $\tilde{T}_n(\theta_0)$ and {\color{black} $V_n(\theta_0)$}.
\end{remark}
\begin{discuss}
{\color{black}}% {{\color[rgb]{0.8,0,0} $\tilde{T}_n\approx T_n$よりLAMNの1が成り立ち，
\begin{equation*}
(W_n,\tilde{T}_n)\approx (T_n^{-1/2}1_{\{T_n(\theta_0) \ {\rm is \ p.d.}\}}T_n^{1/2}W_n,T_n)\to (T^{-1/2}T^{1/2}W,T)=(W,T)
\end{equation*}
よりOK.
}
\end{discuss}


Let $(m_n)_{n=1}^\infty$ be a sequence of positive integers.
Let $\{{\color{black} \mathfrak{R}_{n,j}}\}_{j=1}^{m_n}$ be a sequence of complete, separable metric spaces,
 and let $\Theta$ be an open subset of $\mathbb{R}^d$.
Let {\color{black} $\mathfrak{R}_n=\mathfrak{R}_{n,1}\times \cdots \times \mathfrak{R}_{n,m_n}$}.
We consider statistical experiments {\color{black} $(\mathfrak{R}_n, \mathcal{B}(\mathfrak{R}_n), \{P_{\theta,n}\}_{\theta\in\Theta})$}.
{\color{black}}% {\color[rgb]{0,0,0.8} Let $X_j = X_{n,j} : {\color{black} \mathfrak{R}_n \to \mathfrak{R}_{n,j}}$ be the natural
projection, $\bar{X}_j = \bar{X}_{n,j} = (X_1, \dots, X_j)$, and
$\mathcal{F}_j = \mathcal{F}_{n,j}=\sigma(\bar{X}_j)$ 
for $0\leq j\leq m_n$.
Suppose that there exists a $\sigma$-finite measure $\mu_j = \mu_{n,j}$
on {\color{black} $\mathfrak{R}_{n,j}$}
such that $P_{\theta,n}(X_1 \in \cdot) \ll \mu_1$ and 
$P_{\theta,n}(X_j \in \cdot | \bar{X}_{j-1} = \bar{x}_{j-1})\ll \mu_j$ for 
all $\bar{x}_{j-1} \in {\color{black} \mathfrak{R}_{n,1} \times \dots \times \mathfrak{R}_{n,j-1}}$,
$2\leq j\leq m_n$.
}
\begin{discuss}
{\color{black}}% {{\color[rgb]{0.8,0,0} $P_{\theta,n}|_{\mathcal{F}_k}\ll \mu_{n,1}\otimes \cdots \otimes \mu_{n,k}$ for $1\leq k\leq m_n$なら
\begin{equation*}
F_k(\bar{x}_k)=\frac{dP_{\theta,n}|_{\mathcal{F}_k}}{d(\mu_{n,1}\otimes \cdots \otimes \mu_{n,k})}
\end{equation*}
として
\begin{equation*}
P_{\theta,n}(x_j\in A|\bar{x}_{j-1})=\int_A\frac{F_j(\bar{x}_j)}{F_{j-1}(\bar{x}_{j-1})}\mu_j(dx_j)
\end{equation*}
か．
}
\end{discuss}
Let $E_\theta = E_{\theta,n}$ denote the expectation with respect to
$P_{\theta,n}$, and let $p_j = p_{n,j}$ be the conditional density functions defined by
\begin{equation*}
p_1(\theta)= \frac{dP_{\theta,n}(X_1\in
 \cdot)}{d\mu_1} : {\color{black} \mathfrak{R}_{n,1}} \to \mathbb{R}, 
\quad p_j(\theta)= 
\frac{dP_{\theta,n}(X_j\in \cdot | \bar{X}_{j-1})}{d\mu_{j}} : 
{\color{black} \mathfrak{R}_{n,j}} \to \mathbb{R}
\end{equation*}
for $2\leq j\leq m_n$.
Then we can see that for $g : {\color{black} \mathfrak{R}_{n,1} \times \dots \mathfrak{R}_{n,j}}
\to \mathbb{R}$,
\begin{equation}\label{condE-eq}
\int_{{\color{black} \mathfrak{R}_{n,j}}} p_j(\theta)g(\bar{X}_{j-1}, \cdot)d\mu_{j}
=E_{\theta}[g(\bar{X}_{j-1}, X_{j})|\mathcal{F}_{j-1}].
\end{equation}

\begin{discuss}
{\color{black}}% {{\color[rgb]{0.8,0,0} ∵任意のBorel functions $g,h$に対して
\begin{equation*}
E_{\theta}[\int p_j(\theta)g(x_j,\bar{x}_{j-1})d\mu_jh(\bar{x}_{j-1})]=E_{\theta}[g(x_j,\bar{x}_{j-1})h(\bar{x}_{j-1})]
\end{equation*}
だから．
}
\end{discuss}


\begin{description}
\item[Assumption (A1).] There {\color{black} is a} $d\times d$ positive definite matrix $r_n$
and {\color{black} measurable} functions 
$$ {\color{black} \dot{\xi}_{n,j}(\theta_0, \cdot ): \mathfrak{R}_{n,1}\times \dots \times \mathfrak{R}_{n,j} \to \mathbb{R}^d} $$
 such that for every $h\in\mathbb{R}^d$,
\begin{equation}\label{L2-regu-eq}
\sum_{j=1}^{m_n}E_{\theta_0}\bigg[\int
 {\color{black}}% {\color[rgb]{0,0,0.8} [\xi_{n,j}}(\theta_0,h)-\frac{1}{2}h^{\top}{\color{black}}% {\color[rgb]{0,0,0.8}
 r_n}{\color{black}}% {\color[rgb]{0,0,0.8} \dot{\xi}_{j}}(\theta_0)]^2d {\color{black}}% {\color[rgb]{0,0,0.8} \mu_{j}}\bigg]\to 0
\end{equation}
as $n\to\infty$, where
\begin{equation*}
{\color{black}}% {\color[rgb]{0,0,0.8}\xi_{n,j}}(\theta_0,h)=\sqrt{p_j(\theta_0+{\color{black}}% {\color[rgb]{0,0,0.8} r_nh})}-\sqrt{p_j(\theta_0)}
\end{equation*}
and
  $${\color{black} \dot{\xi}_{j}(\theta_0)= \dot{\xi}_{n,j}(\theta_0, \bar{X}_{j-1},\cdot ): 
    \mathfrak{R}_{n,j} \to \mathbb{R}^d.} $$
\end{description}
Condition~(A1) {\color{black} is} the $L^2$ regularity {\color{black} condition}.

For a vector $x=(x_1,\cdots, x_k)$, we denote
$\partial_x^l=(\frac{\partial^l}{\partial x_{i_1}\cdots \partial
x_{i_l}})_{i_1,\cdots,i_l=1}^k$.
If $p_j$ is smooth with respect to $\theta$ and $p_j\neq 0$, then the log-likelihood ratio is rewritten as
\begin{equation*}
\log \frac{dP_{\theta',n}}{dP_{\theta,n}}=\sum_{j=1}^{m_n}\log\frac{p_j(\theta')}{p_j(\theta)}
=\sum_{j=1}^{m_n}\int^1_0\frac{\partial_{\theta}p_j}{p_j}(t\theta'+(1-t)\theta)dt(\theta'-\theta).
\end{equation*}
To show the LAMN property, we {\color{black} must} identify the limit distribution of this function under $P_{\theta,n}$.
{\color{black} Doing so} requires estimates for density ratios with different probability measures, which are not easy {\color{black} to obtain} for stochastic processes in general.
Gobet~\cite{gob01} {\color{black}}% {\color[rgb]{0,0,0.8} dealt} with this problem for discretely observed diffusion processes 
by using estimates from below and above by Gaussian density functions and show the LAMN property of that model.
\begin{discuss}
{\color{black}}% {{\color[rgb]{0.8,0,0} 
\begin{eqnarray}
P_{\theta,n}(A_1\times \cdots \times A_{m_n})&=&E_\theta[E[1_{A_{m_n}}|\bar{x}_{m_n-1}]1_{A_1\times \cdots A_{m_n}}]
=E_\theta[\int 1_{A_{m_n}}p_{m_n}(\theta)d\mu_{n,m_n}1_{A_1\times \cdots A_{m_n-1}}] \nonumber \\
&=&E_\theta[\int \int 1_{A_{m_n-1}\times A_{m_n}}p_{m_n-1}p_{m_n}d\mu_{n,m_n}d\mu_{n,m_n-1}1_{A_1\times \cdots \times A_{m_n-2}}]=\cdots  \nonumber \\
&=&\int \cdots \int 1_A\prod_jp_j\prod_jd\mu_j. \nonumber
\end{eqnarray}
よって
\begin{equation*}
\frac{dP_{\theta,n}}{d\otimes_j\mu_j}=\prod_jp_j.
\end{equation*}
ここでは条件付き確率が$p_j$でその積でdensityが書けるのは一般に成立．あとはtransition densityの表現で問題が起きないかどうか．
}
\end{discuss}


On the other hand, if $p_j\in C^2(\Theta)$ and $p_j(\theta_t)\neq 0$ for any $t\in [0,1]$ $\mu_j$-a.e.
for $\theta_t=\theta_0+t{\color{black}}% {\color[rgb]{0,0,0.8} r_n}h$, then by setting
${\color{black}}% {\color[rgb]{0,0,0.8}\dot{\xi}_{j} (\theta_0)}=\partial_{\theta}p_j(\theta_0)p_j(\theta_0)^{-1/2}$ we obtain
\EQN{\label{L2-regu-equiv}
&\int
 [{\color{black}}% {\color[rgb]{0,0,0.8}\xi_{n,j}}(\theta_0,h)-\frac{1}{2}h^{\top}{\color{black}}% {\color[rgb]{0,0,0.8} r_n \dot{\xi}_{j} (\theta_0)}]^2d{\color{black}}% {\color[rgb]{0,0,0.8}
 \mu_{j}} \\
&\quad =\int \bigg[h^{\top}{\color{black}}% {\color[rgb]{0,0,0.8}
 r_n}\int^1_0\frac{\partial_{\theta}p_j(\theta_t)}{2\sqrt{p_j(\theta_t)}}dt-\frac{1}{2}h^{\top}{\color{black}}% {\color[rgb]{0,0,0.8}
 r_n}\frac{\partial_{\theta}p_j(\theta_0)}{\sqrt{p_j(\theta_0)}}\bigg]^2d{\color{black}}% {\color[rgb]{0,0,0.8}
 \mu_{j}} \\
&\quad =\int \bigg[\frac{h^{\top}{\color{black}}% {\color[rgb]{0,0,0.8}
 r_n}}{2}\int^1_0\int^t_0\bigg(\frac{\partial_{\theta}^2p_j(\theta_s)}{\sqrt{p_j(\theta_s)}}-{\color{black} \frac{\partial_{\theta}p_j(\partial_{\theta}p_j)^\top}{2p_j^{3/2}}(\theta_s)}\bigg)dsdt{\color{black}}% {\color[rgb]{0,0,0.8}
 r_n}h\bigg]^2d{\color{black}}% {\color[rgb]{0,0,0.8}
 \mu_{j}} \\
&\quad \leq \frac{1}{4}\sup_{0\leq s\leq
 1}E_{\theta_s}\bigg[\bigg\{h^{\top}{\color{black}}% {\color[rgb]{0,0,0.8}
 r_n}\bigg(\frac{\partial_{\theta}^2p_j(\theta_s)}{p_j(\theta_s)}-{\color{black} \frac{\partial_{\theta}p_j(\partial_{\theta}p_j)^\top}{2p_j^2}(\theta_s)}\bigg){\color{black}}% {\color[rgb]{0,0,0.8}
 r_n}h\bigg\}^2\bigg|{\color{black}}% {\color[rgb]{0,0,0.8}\mathcal{F}_{j-1}}\bigg].
}
\begin{discuss}
{\color{black}}% {{\color[rgb]{0.8,0,0} Ibragimov Hasminskiiの教科書のII.3のRemark 3.2でL2 regularityを上のような二階微分の条件に変換している．
$p$は$(U_0^{n,\theta},\cdots, U_{k_j-k_{j-1}}^{n,\theta})$のdensity. 
}
\end{discuss}
In the right-hand side of the above inequality, 
{\color{black} the value $\theta_s$ of the parameter is the same for the probability measure of expectation and $p_j$ in the integrand, and therefore we do not need estimates for the transition density ratios.}

Thus, a scheme with the $L^2$ regularity condition does not require estimates for {\color{black} the} transition density function.
This is a big advantage, and this scheme {\color{black} can be} applicable to multi-dimensional integrated diffusion processes
 {\color{black} and  degenerate} diffusion processes,
where it is difficult to obtain estimates for transition density functions.

Define
{\color{black}}% {\color[rgb]{0,0,0.8}
\begin{equation*}
\eta_j= \left(\frac{\dot{\xi}_{j}(\theta_0)}{\sqrt{p_j(\theta_0)}} 1_{\{p_j(\theta_0)
 \neq 0\}}\right) (X_j).
\end{equation*}
}
\begin{description}
\item[Assumption (A2).] 
$E_{\theta_0}[|\eta_j|^2|\mathcal{F}_{j-1}] < \infty $ and 
$E_{\theta_0}[\eta_j|\mathcal{F}_{j-1}]=0 $, $P_{\theta_0,n}$-almost
	   surely 
for every $j\geq 1$. 

\item[Assumption (A3).] For every $\epsilon >0$ and $h\in \mathbb{R}^d$,
\begin{equation*}
\sum_{j=1}^{m_n}E_{\theta_0}[|h^{\top}{\color{black}}% {\color[rgb]{0,0,0.8} r_n}\eta_j|^21_{\{|h^{\top}{\color{black}}% {\color[rgb]{0,0,0.8} r_n}\eta_j|>\epsilon\}}]\to 0.
\end{equation*}
\item[Assumption (A4).] For every $h\in\mathbb{R}^d$, there exists a constant $K>0$ such that
\begin{equation*}
\sup_{n\geq 1}\sum_{j=1}^{m_n}E_{\theta_0}[|h^{\top}{\color{black}}% {\color[rgb]{0,0,0.8} r_n}\eta_j|^2]\leq K.
\end{equation*}
\end{description} 
Let 
\begin{equation}\label{TnWn-def}
{\color{black} T_n}={\color{black}}% {\color[rgb]{0,0,0.8}
r_n}\sum_{j=1}^{m_n}E_{\theta_0}[\eta_j\eta_j^{\top}|{\color{black}}% {\color[rgb]{0,0,0.8} \mathcal{F}_{j-1}}]{\color{black}}% {\color[rgb]{0,0,0.8} r_n}
\quad {\rm and} \quad {\color{black} V_n}={\color{black}}% {\color[rgb]{0,0,0.8} r_n}\sum_{j=1}^{m_n}\eta_j.
\end{equation}
\begin{description}
\item[Assumption (A5).] 
There exists a random $d\times d$ symmetric matrix $T$ such that
$P[T \ {\rm is \ {\color{black} n.d.}}]=1$ and
\begin{equation*}
\mathcal{L}(({\color{black} V_n}, T_n)|P_{\theta_0,n}) \to  \mathcal{L}(T^{1/2}W,T),
\end{equation*}
where $W \sim N(0,I_d)\perp T$. 
{\color{black} \item[Assumption (P).] $T$ in (A5) is positive definite almost surely.}
\end{description} 
\begin{discuss}
{\color{black}}% {{\color[rgb]{0.8,0,0} (A2)を使っているところは今のところJeg82 Lem 1とLem7の証明．ちなみにLem1はLem 5,6でも使う．}
\end{discuss}

Conditions (A1)--(A4) correspond to (2.A.1), (2.A.2), (2.A.4), {\color{black} and} (2.A.5) in~\cite{jeg82}, respectively.
{\color{black} Condition (A5) ensures Point~2 of Condition (L)}. For the sequential observations in~\cite{jeg82},
convergence of ${\color{black}}% {\color[rgb]{0,0,0.8} r_n}\sum_{j=1}^{m_n}\eta_j$ always holds by {\color{black} virtue} of Hall~\cite{hal77} (see (2.3) in~\cite{jeg82}).
However, the results of~\cite{hal77} cannot be applied to the triangular array observations,
so we {\color{black} instead} assume (A5) for our scheme.
{\color{black} To check (A5), the results in Sweeting~\cite{swe80} are useful. Moreover, it}
is {\color{black}}% {\color[rgb]{0,0,0.8} not} difficult to check (A5) for statistical models of discretely observed diffusion processes
{\color{black} by using a martingale central limit theorem}.
See, for example, {\color{black} Theorems~\ref{degenerate-LAMN-thm} and~\ref{partial-LAMN-thm} and their proofs}.

\begin{discuss}
{\color{black}}% {{\color[rgb]{0.8,0,0} Sweetingは$\sum_j\partial_\theta(\partial_\theta p_j/p_j)\overset{P}\to \Gamma_\theta$ for any $\theta$ならだいたいOKか}
\end{discuss}

\begin{theorem}\label{L2regu-thm}
Assume $(A1)$--$(A5)$. {\color{black} Then (L) holds true with $T_n$ and $V_n$ in~(\ref{TnWn-def}) for the family $\{P_{\theta,n}\}_{\theta,n}$ of probability measures.
If further (P) is satisfied, {\color{black} then} $\{P_{\theta,n}\}_{\theta,n}$ satisfies the LAMN condition at $\theta=\theta_0$
with {\color{black} $\tilde{T}_n$} in Remark~\ref{Gamma-pd-rem}.}
\end{theorem}

\begin{arXiv}
A proof is given in the appendix.
\end{arXiv}
\begin{as}
{\color{black} The proof is {\color{black} given in} Section~\ref{L2regu-proof-section} {\color{black} of} the supplementary {\color{black} material}~\cite{fuk-ogi20s}.}
\end{as}

\begin{remark}
We assumed that ${\color{black}}% {\color[rgb]{0,0,0.8} r_n}$ is positive definite because {\color{black} this assumption is made} in the definition of the LAMN property in Jeganathan~\cite{jeg82} (Definition~1).
However, we can see that Theorem~\ref{L2regu-thm} holds even {\color{black} if} ${\color{black}}% {\color[rgb]{0,0,0.8} r_n}$ is a nondegenerate {\color{black} asymmetric} matrix.
In that case, {\color{black} even} though the assumptions of convolution theorem (Corollary 1) in~\cite{jeg82} are not satisfied,
the convolution theorem in H\'ajek~\cite{haj70} is satisfied when {\it local asymptotic normality} is satisfied
({\color{black} i.e.,} $T$ in (A5) is non-random) and the operator norm of ${\color{black}}% {\color[rgb]{0,0,0.8} r_n}{\color{black}}% {\color[rgb]{0,0,0.8} r_n^\top}$ converges to zero.
\end{remark}









\subsection{The LAMN {\color{black} property} via Malliavin calculus techniques}\label{LAMN-Malliavin-subsection}

\begin{discuss}
{\color{black}}% {{\color[rgb]{0.8,0,0} adaptive推定をするわけではないから$\Theta_1\times \Theta_2$型でやるかどうかは関係ない}
\end{discuss}

Gobet~\cite{gob01,gob02} used Malliavin calculus techniques to show the LAMN property for discretely observed diffusion processes.
Gloter and Gobet~\cite{glo-gob08} developed Gobet's scheme {\color{black} into a} more general one and showed the LAMN property for a one-dimensional integrated diffusion process.
{\color{black}}% {\color[rgb]{0,0,0.8} These approaches require estimates for transition density
functions by Gauss density functions, {\color{black} thereby} hampers
multi-dimensional extension {\color{black} of their results}. Our alternative approach
 introduces} tractable sufficient conditions to show the LAMN property for smooth observations in the Malliavin sense,
by combining with a scheme with the $L^2$ regularity condition in Section~\ref{L2regu-subsection}.  
In particular, we can show the $L^2$ regularity condition under (B1) and (B2), 
which are related to {\color{black} the} smoothness of observations and estimates for the inverse of {\color{black} the} Malliavin matrix (Theorem~\ref{Malliavin-LAMN-thm2}).
If further, observations have {\color{black} a} Gaussian approximation like {\color{black} the Euler--Maruyama} approximation, {\color{black} then the} sufficient conditions for the LAMN property are simplified {\color{black} as} in Theorem~\ref{Malliavin-LAMN-thm}.
\begin{discuss}
{\color{black}}% {{\color[rgb]{0.8,0,0} Jump Malliavinを含むJacodの教科書のMalliavin operatorを使ったアプローチは$\partial p/p$の表現は得られそうだが，
その後の証明を丸々焼き直す必要があるので今回はやめておく．
integration-by-parts-settingを使った議論も同じ．
非有界作用素の理論はB→Hの作用素をあまり扱っていないから辞めといた方がよさそう．}
\end{discuss}



\begin{discuss}
{\color{black}}% {{\color[rgb]{0.8,0,0} Let $(B,H,\mu)$ be an abstract Wiener space.}
\end{discuss}

We assume that {\color{black} $\Theta$ is convex and that} $m_n\to \infty$ as $n\to \infty$.
\begin{discuss}
{\color{black}}% {{\color[rgb]{0.8,0,0} Proposition 3.1で$C_0^\infty$の可分性を言うためにconvexが必要か．不要かもしれないが，$\theta_0$固定のLAMNでは$\Theta$は凸で問題ない}
\end{discuss}
Let $(\epsilon_n)_{n=1}^\infty$ be a sequence of positive numbers and $(\Omega,\mathcal{F},P)$ be a probability space. 
Let $({\color{black}}% {\color[rgb]{0,0,0.8} k^n_j})_{j=0}^{m_n}$ be an increasing sequence of
nonnegative integers such that ${\color{black}}% {\color[rgb]{0,0,0.8} k^n_0}=0$.
{\color{black} Hereinafter, we abbreviate $k^n_j$ as simply $k_j$.}
Let $N_n=k_{m_n}$ and $X_j^{n,\theta}$ be an $\mathbb{R}^{k_j-k_{j-1}}$-valued random variable on $(\Omega,\mathcal{F},P)$ for $1\leq j\leq m_n$.
Let $P_{\theta,n}$ be the induced probability measure by $\{X_j^{n,\theta}\}_{j=1}^{m_n}$ on $(\mathbb{R}^{N_n},\mathcal{B}(\mathbb{R}^{N_n}))$ and $\mathcal{F}_{j,n}=\{A\times \mathbb{R}^{N_n-k_j}|A\in \mathcal{B}(\mathbb{R}^{k_j})\}\subset \mathcal{B}(\mathbb{R}^{N_n})$.
\begin{discuss}
{\color{black}}% {{\color[rgb]{0.8,0,0} (B4)を言うために$(\Omega,\mathcal{F},P)$の存在が必要．$P_{\theta,n}$はデータだけの空間にしたいからそこで$\mathcal{L}(\tilde{F})$は定義したくない．}
\end{discuss}
For each $1\leq j\leq m_n$, we adopt the notation of Nualart~\cite{nua06}. {\color{black} Specifically,} let $H_j$ be a real separable Hilbert space and
$W_j=\{W_j(h),h\in H_j\}$ be an isonormal Gaussian process defined on a complete probability space $(\Omega_j,\mathcal{G}_j,Q_j)$.
We assume that $\mathcal{G}_j$ is generated by $W_j$. {\color{black} Even though} these
objects possibly depend on $n$, we omit the dependence in our notation.
\begin{discuss}
{\color{black}}% {{\color[rgb]{0.8,0,0} 元の定義：For each $1\leq j\leq m_n$, let $H_j$ be a real separable Hilbert space, $B_j$ be a real separable Banach space,
$\mathcal{B}_j$ be the Borel $\sigma$-field on $B_j$ and $Q_j$ be a probability measure such that
$(B_j,\mathcal{B}_j,Q_j)$ be a complete probability space.
Let $W_j=\{W_j(h),h\in H_j\}$ be isonormal Gaussian process defined on $(B_j,\mathcal{B}_j,Q_j)$,
and we assume that $\mathcal{B}_j$ is generated by $W_j$. $B_j=L^p(\Omega,\mathcal{G},Q)$ということ．
}
\end{discuss}
Let $\delta_j$ be {\color{black} the Hitsuda--Skorokhod integral (the divergence operator), $D_j$ be the Malliavin--Shigekawa} derivative, 
and $\mathcal{S}_j=\{f(W_j(h_1),\cdots,W_j(h_k));k\in\mathbb{N},h_i\in H_j \ (i=1,\cdots,k), f\in C^\infty(\mathbb{R}^k)\}$.
For $k\in\mathbb{Z}_+$ and $p\geq 1$, $\lVert \cdot \rVert_{k,p}$ denotes the operator on $\mathcal{S}_j$ defined by
\begin{equation*}
\lVert F\rVert_{k,p}=\bigg[E_j[|F|^p]+\sum_{l=1}^kE_j[\lVert D_j^lF\rVert_{H_j^{\otimes l}}^p]\bigg]^{1/p},
\end{equation*}
where $E_j$ denotes the expectation with respect to $Q_j$. 
\begin{discuss}
{\color{black}}% {{\color[rgb]{0.8,0,0} ミンコフスキーの不等式から劣加法性が成立し，ノルムになる．ソボレフ空間のノルムの定義の仕方と一致．}
\end{discuss}
Let $\mathbb{D}^{k,p}_j$ be the completion of $\mathcal{S}_j$ with respect to the distance $d(F,G):=\lVert F-G \rVert_{k,p}$.
For general properties of $W_j$, $D_j$, and $\delta_j$, {\color{black} see} Nualart~\cite{nua06}.
Let {\color{black} $F_{n,\theta,j,\bar{x}_{j-1}}$} be an $\mathbb{R}^{k_j-k_{j-1}}$-valued random variable on $(\Omega_j,\mathcal{G}_j)$
such that
$Q_jF_{n,\theta,j,\bar{x}_{j-1}}^{-1}={\color{black}}% {\color[rgb]{0,0,0.8} P}(X_j^{n,\theta}\in
\cdot |\bar{X}_{j-1}^{n,\theta}=\bar{x}_{j-1})$, where
$\bar{X}_{j-1}^{n,\theta}=\{ {\color{black}}% {\color[rgb]{0,0,0.8} X_l^{n,\theta}}\}_{l=1}^{j-1}$ {\color{black}}% {\color[rgb]{0,0,0.8} and $\bar{x}_{j-1}\in \mathbb{R}^{k_{j-1}}$}.
We assume that $F_{n,\theta,j,\bar{x}_{j-1}}$ is Fr\'echet differentiable with respect to $\theta$ on $L^p(\Omega_j)$ for any $p>1$
and denote its derivative by $\partial_\theta F_{n,\theta,j,\bar{x}_{j-1}}=(\partial_{\theta_1} F_{n,\theta,j,\bar{x}_{j-1}},\cdots, \partial_{\theta_d} F_{n,\theta,j,\bar{x}_{j-1}})^\top$.
We often omit the parameter $\bar{x}_{j-1}$ in $F_{n,\theta,j,\bar{x}_{j-1}}$ and write $F_{n,\theta,j}$.
Let $C^\infty_o$ denote the space of all $C^\infty$ functions with compact {\color{black} support}. 
For a matrix $A$, we denote its {\color{black} element $(i,j)$} by $[A]_{ij}$. 
{\color{black} Similarly, we denote by $[V]_l$ the $l$-th element of a vector $V$.}
Let $\bar{k}_n=\max_j(k_j-k_{j-1})$.
\begin{discuss}
{\color{black}}% {{\color[rgb]{0.8,0,0} 積分観測では$m_n\approx n$}
\end{discuss}

We assume the following conditions.
\begin{description}
\item[Assumption (B1).]  $\partial_\theta^l{\color{black} [F_{n,\theta,j}]_i}\in \cap_{p>1}\mathbb{D}^{4-l,p}_j$ for any $n,\theta,j,i,0\leq l\leq 3$, and $\sup_{n,i,j,\bar{x}_{j-1},\theta}\lVert \partial_\theta^l{\color{black} [F_{n,\theta,j}]_i}\rVert_{4-l,p}<\infty$ for $p>1$.
\begin{discuss}
{\color{black}}% {{\color[rgb]{0.8,0,0} $K^{-1}\in \mathbb{D}^{1,2}$等のために$F$の$p$はかなり大きくとる必要があるから任意の$p$でよい．(B3)の$p$も同じ}
\end{discuss}
\item[Assumption (B2).]  The matrix $K_j(\theta)=(\langle {\color{black} D_j[F_{n,\theta,j}]_k,D_j[F_{n,\theta,j}]_l}\rangle_{H_j})_{k,l}$ is invertible almost surely for any $j,\bar{x}_{j-1}$ and $\theta$, and there exists a constant $\alpha_n\geq 1$ such that
\begin{equation*}
\sup_{i,l,j,\bar{x}_{j-1},\theta}\lVert [K_j^{-1}(\theta)]_{il}\rVert_{2,8}\leq \alpha_n, \quad {\rm and} \quad \epsilon_n^2\bar{k}_n^4\sqrt{m_n}\alpha_n^2\to 0
\end{equation*}
as $n\to\infty$.
\end{description}
\begin{discuss}
{\color{black}}% {{\color[rgb]{0.8,0,0} 
$\lVert \cdot \rVert_{k,p}$の$p$に関する単調性は，$q>p$の時
\begin{eqnarray}
\lVert F\rVert_{k,p}\leq \bigg[E_j[|F|^q]^{p/q}+\sum_{l=1}^kE_j[\lVert D_j^l F\rVert_{H^{\otimes l}}^q]^{p/q}\bigg]^{1/p}
\leq \lVert F\rVert_{k,q}. \nonumber
\end{eqnarray}
($x,y\geq 0$, $r\in [0, 1]$に対して，$(x+y)^r\leq x^r+y^r)$)
よりわかるので，(B2)の定義は問題なし．

$(\det K_j)^{-1}$の評価より$[K_j^{-1}]_{il}$の評価が以下では必要なのでそれを仮定する．$(\det K_j)^{-1}$評価からも得られるが余因子が絡んできて行列サイズが発散しうる状況で一般的に書くのは面倒．

＜(B2)の有界性を使った評価に関して＞\\
今までは$a,b$の有界性などから定数は初期値$\bar{x}_{j-1}$に依存しない形だったから$\sup_{\bar{x}_{j-1}}E\j[\cdot]$が評価できているので$E[\cdot]$も評価できる．
エルゴードの時もモーメント評価ができる設定位を考えればよいから(B2)は$E$評価でいいだろう．しかしエルゴードを考えると$\mathcal{S}(c_n)$を使う方が良い．
For a sequence $(c_n)_{n=1}^\infty$ of positive numbers, $\mathcal{S}(c_n)$ denotes the set of all sequences $(p_j(\bar{x}_{j-1},\theta))_{j=1}^{m_n}$ of measurable functions such that
\begin{equation*}
\sup_nc_n^{-1}E\bigg[\sum_{j=1}^{m_n}|p_j(\bar{x}_{j-1})|\bigg]<\infty.
\end{equation*}

densityの存在を言うのにマルコフ性の仮定がなくてもおそらく大丈夫か．
多分$P(\cdot |\bar{x}_{j-1})=Q_jF^{-1}$と書けることと$K$が計算できて$K^{-1}$が評価できることが実質的にマルコフ並みの良さを仮定しているか．
$x_1,\cdots,x_{j-1}$に複雑に依存していたらこんなにきれいに書けないだろう．
}
\end{discuss}

Let $\theta_0$ be the true value of parameter $\theta$. 
We will see later in Proposition~\ref{delLogP-eq} that $F_{n,\theta,j}$ admits a density $p_{j,\bar{x}_{j-1}}(x_j,\theta)$
{\color{black} that} satisfies $p_{j,\bar{x}_{j-1}}(x_j,\cdot) \in C^2(\Theta)$ almost {\color{black} everywhere} in $x_j\in \mathbb{R}^{k_j-k_{j-1}}$ under (B1) and (B2).
{\color{black}}% {\color[rgb]{0.8,0.3,0} Let $N_j=\{x_j\in\mathbb{R}^{k_j-k_{j-1}}|\sup_{\theta\in\Theta}p_{j,\bar{x}_{j-1}}(x_j,\theta)>\inf_{\theta\in\Theta}p_{j,\bar{x}_{j-1}}(x_j,\theta)=0\}$
and $M_j=\{x_j\in\mathbb{R}^{k_j-k_{j-1}}|\inf_{\theta\in\Theta}p_{j,\bar{x}_{j-1}}(x_j,\theta)>0\}$. We further assume the following condition.
\begin{description}
\item[Assumption (N1).] For any $h\in \mathbb{R}^d$, 
\begin{equation*}
E_{\theta_0}\bigg[\sum_{j=1}^{m_n}\int_{N_j}p_{j,\bar{x}_{j-1}}(x_j,\theta_0+r_nh)dx_j\bigg]\to 0
\end{equation*}
as $n\to\infty$.
\end{description}
}

{\color{black}}% {\color[rgb]{0.8,0.3,0} If $\sup_{\theta\in\Theta}p_j(x_j,\theta)=0$ or $\inf_{\theta\in\Theta}p_j(x_j,\theta)>0$, we have $x_j\in N_j^c$.
Condition~(N1) says that the probability of other cases is asymptotically negligible. This condition is used to validate an estimate such as (\ref{L2-regu-equiv}).
However, if $F_{n,\theta,j}$ is approximated by a Gaussian random variable and satisfies (B3) and (N2) below,
{\color{black} then} we can check (A1) without (N1) (see Lemma~\ref{L2-regu-lemma}).}

With these definitions, the following theorem shows that the $L^2$ regularity condition is automatically satisfied under (B1), (B2), and (N1).
{\color{black}}% {\color[rgb]{0.8,0.3,0} Let 
\begin{equation}\label{N1-xi-def}
\dot{\xi}_j(\theta)=\frac{\partial_\theta p_j}{2\sqrt{p_j}}1_{M_j}(x_j,\theta),
\quad \eta_j=\frac{\partial_\theta p_j}{2p_j}1_{M_j}(x_j,\theta_0).
\end{equation}
}
\begin{theorem}\label{Malliavin-LAMN-thm2}
Assume (B1), (B2), (N1), (A4), and (A5) {\color{black}}% {\color[rgb]{0.8,0.3,0} with $\dot{\xi}_j(\theta)$ and $\eta_j$ defined in (\ref{N1-xi-def})}. 
{\color{black} Then (L) holds true for $\{P_{\theta,n}\}_{\theta,n}$ at $\theta=\theta_0$ with $r_n = \epsilon_nI_d$.
If further (P) is satisfied, {\color{black} then} $\{P_{\theta,n}\}_{\theta,n}$ satisfies the LAMN condition at $\theta=\theta_0$.}
\end{theorem}


In the following, we give tractable sufficient conditions for (A4) and (A5) when $F_{n,\theta,j}$ has a Gaussian approximation $\tilde{F}_{n,\theta,j}$.
\begin{description}
\item[Assumption (B3).]  {\color{black} There exist a matrix $B_{j,i,\theta}=B_{j,i,\theta,\bar{x}_{j-1},n}$ and $h_{j,l}=h_{n,\theta,j,l,\bar{x}_{j-1}}\in H_j \ (1\leq l\leq k_j-k_{j-1})$
such that $\tilde{F}_{n,\theta,j,\bar{x}_{j-1}}=(W_j(h_{j,l}))_{l=1}^{k_j-k_{j-1}}$ is Fr\'echet differentiable with respect to $\theta$ on $L^p$ space,
and $\partial_{\theta_i}\tilde{F}_{n,\theta,j}=B_{j,i,\theta}\tilde{F}_{n,\theta,j}$.
Moreover, $\partial_\theta B_{j,i,\theta}$ exists and is continuous with respect to $\theta$, and there exists a constant $C_p$ and a sequence $(\rho_n)_{n\in\mathbb{N}}$ of positive numbers such that
}
\begin{equation*}
\sup_{{\color{black}}% {\color[rgb]{0.8,0.3,0} n,}i,j,\bar{x}_{j-1},\theta,l_1,l_2}|[{\color{black}}% {\color[rgb]{0.8,0.3,0} \partial_\theta^l}B_{j,i,\theta}]_{l_1,l_2}|<\infty,
\end{equation*}
{\color{black} and
\EQQ{\lVert [F_{n,\theta,j}-\tilde{F}_{n,\theta,j}]_{i'}\rVert_{3,p} 
+\lVert \partial_{\theta_i}[F_{n,\theta,j}-\tilde{F}_{n,\theta,j}]_{i'}\rVert_{2,p} \leq C_p\rho_n}}
for {\color{black} $p>1$}, $1\leq j\leq m_n$, $1\leq i\leq d$, $1\leq i'\leq k_j-k_{j-1}$, $\theta\in \Theta$ and $\bar{x}_{j-1}$.
\begin{discuss}
{\color{black}}% {{\color[rgb]{0.8,0,0} $X$から$W$を復元するには2.3の$a$: invertibleは必要．復元しないと条件付き期待値を外すのが難しい．

$\delta D\tilde{F}=\tilde{F}$も必要だが，この時$E[F]=E[\delta DF]=0$で，Nualart Prop 1.2.2より
$E[\lVert DF\rVert^2_H]=\sum_{n=1}^\infty n\lVert J_nF\rVert_2^2<\infty$.
\begin{equation*}
E[\lVert DF\rVert^2_H]=E[F\delta DF]=E[F^2]=\sum_{n=1}^\infty \lVert J_nF\rVert^2_2
\end{equation*}
より$J_nF=0$ for $n\geq 2$なのである$h\in H$があって$F=W(h)$. $\delta D\tilde{F}=c\tilde{F}$も考えられるが，$\tilde{K}$がdeterministicになるには$F=W(h)$が必要だからこれを仮定する．}
\end{discuss}
\end{description}
\begin{discuss}
{\color{black}}% {{\color[rgb]{0.8,0,0} 条件付き期待値をはずすにはこういう評価が必要．
densityの存在はマリアバンマトリックスの非退化性とNualart Thm 2.1.2からでる．
$\tilde{K}$の各要素の評価は$K$の評価と$K-\tilde{K}$の評価から出る．
}
\end{discuss}

{\color{black}}% {\color[rgb]{0.8,0.3,0} For a statistical model of diffusion processes, $F_{n,\theta,j}$ corresponds to {\color{black} normalized} discrete observations and $\tilde{F}_{n,\theta,j}$ corresponds their {\color{black} Euler--Maruyama} approximations.
See (\ref{F-def}) and (\ref{tildeF-def}) for an example.}

Let $\tilde{K}_j(\theta)=(\langle h_{j,l_1},h_{j,_2}\rangle_{H_j})_{l_1,l_2}$.
Then, we will see that for sufficiently large $n$, $\tilde{K}_j(\theta)$ is invertible almost surely under (B1)--(B3) and that $\alpha_n\rho_n\bar{k}_n^2\to 0$ 
in Lemma~\ref{tildeK-est-lemma} of Section~\ref{Mcal-section}.
Let
\begin{equation*}
\mathcal{L}_{j,i,\bar{x}_{j-1}}(u,\theta)=u^\top B_{j,i,\theta}^\top \tilde{K}_j^{-1}(\theta)u-{\rm tr}(B_{j,i,\theta}).
\end{equation*}



Let $\Phi_{j,i}=(B_{j,i,\theta_0}^\top\tilde{K}_j^{-1}(\theta_0)+\tilde{K}_j^{-1}(\theta_0)B_{j,i,\theta_0})/2$, and let
\EQQ{\gamma_j(\bar{x}_{j-1})=(2{\rm tr}(\Phi_{j,i}\tilde{K}_j(\theta_0)\Phi_{j,i'}\tilde{K}_j(\theta_0)))_{i,i'=1}^d.}
\begin{description}
\item[Assumption (B4).] There exist $\mathbb{R}^d$-valued random variables $\{G_j^n\}_{1\leq j\leq m_n,n,\theta}$ and a filtration $\{\mathcal{G}_j\}_{j=1}^{m_n}$ on $(\Omega,\mathcal{F},P)$ 
such that 
  $(X_j^{n,\theta_0},G_j^n)$ is $\mathcal{G}_j$-measurable, $E[G_j^n|\mathcal{G}_{j-1}]=0$, and
  \EQN{\label{Gj-def-eq}
   &Q_j((F_{n,\theta_0,j},(\mathcal{L}_{j,i,\bar{x}_{j-1}}(\tilde{F}_{n,\theta_0,j},{\color{black} \theta_0}))_{i=1}^d)\in A)|_{\bar{x}_{j-1}=\bar{X}_{j-1}^{n,\theta}} \\
   &\quad =P((X_j^{n,\theta_0},G_j^n)\in A|\mathcal{G}_{j-1})
  }
for $A\in \mathcal{B}(\mathbb{R}^{k_j-k_{j-1}}\times \mathbb{R}^d)$ and sufficiently large $n$.
Moreover, 
{\color{black} \EQQ{\sup_n\bigg(\epsilon_n^2\sum_{j=1}^{m_n}E[|\gamma_j(\bar{X}_{j-1}^{n,\theta_0})|]\bigg)<\infty,}} 
\begin{equation}\label{rho-condition}
\alpha_n\rho_n{\color{black}}% {\color[rgb]{0.8,0.3,0} \bar{k}_n^2}\to 0 \quad {\rm and} \quad \epsilon_n^2m_n\alpha_n^3\rho_n{\color{black}}% {\color[rgb]{0.8,0.3,0} \bar{k}_n^6}\to 0
\end{equation} 
as $n\to \infty$, and there exist a random $d\times d$ {\color{black} matrix} $\Gamma$ and a $d$-dimensional standard normal random variable $\mathcal{N}$ such that
\begin{equation}\label{B4-stable-conv}
\bigg(\epsilon_n\sum_{j=1}^{m_n}G_j^n, \epsilon_n^2\sum_{j=1}^{m_n}\gamma_j({\color{black} \bar{X}_{j-1}^{n,\theta_0}})\bigg)\overset{d}\to (\Gamma^{1/2}\mathcal{N},\Gamma).
\end{equation}
\begin{discuss}
{\color{black}}% {{\color[rgb]{0.8,0,0} (\ref{B4-stable-conv})式の十分条件を書こうとすると，特にorthogonal martingaleとの条件付き期待値評価を一般に書くのが大変だからやめておく}
{\color{black}}% {{\color[rgb]{0.8,0,0} 元々$\sup_{i,n}(n^{-1}\sum_{j=1}^{m_n}\sup_{\bar{x}_{j-1}}E_j[\mathcal{L}_{j,i}(\tilde{F}_{n,\theta,j},\theta)^2])<\infty$を仮定していたが，
$\tilde{F}=W(h)\sim N(0,\tilde{K})$より$E_j[\mathcal{L}(\tilde{F})^2]={\rm tr}(B^2)+{\rm tr}(B^\top \tilde{K}^{-1} B\tilde{K})$.
よって
\begin{equation*}
\sup_{i,n}\frac{1}{n}\sum_{j=1}^{m_n}\sup_{\bar{x}_{j-1}}({\rm tr}(B^2)+{\rm tr}(B^\top \tilde{K}^{-1} B\tilde{K}))<\infty
\end{equation*}
を仮定すれば成り立つ．}
\end{discuss}
\item[Assumption (N2).] {\color{black}}% {\color[rgb]{0.8,0.3,0} ${\color{black} [F_{n,\theta,j}]_i}\in \cap_{p>1, r\in\mathbb{N}}\ \mathbb{D}^{r,p}$ for $n,\theta,j,i,\bar{x}_{j-1}$ 
and 
  \EQQ{\sup_{\theta\in\Theta}\lVert {\color{black} [F_{n,\theta,j}]_i}\rVert_{r,p}<\infty} 
for any $n,i,j,\bar{x}_{j-1}$, $r\in\mathbb{N}$ and $p>1$.
{\color{black} Also, 
  \EQQ{\partial_{\theta_i}D_j[\tilde{F}_{n,\theta,j}]_k=\sum_{l=1}^{k_j-k_{j-1}}[B_{j,i,\theta}]_{k,l}D_j[\tilde{F}_{n,\theta,j}]_l}}
and $\sup_{\theta\in\Theta}E_j[|\det K_j^{-1}(\theta)|^p]<\infty$ for any $p>1$, $j$, $k$, $\bar{x}_{j-1}$ and $1\leq i\leq d$.}
\item[Assumption (B5).] {\color{black}}% {\color[rgb]{0.8,0.3,0} (N1) or (N2) holds true.}
{\color{black} \item[Assumption (P$'$).] $\Gamma$ in (B4) is positive definite almost surely.}
\end{description}

{\color{black}
\begin{remark}
{\color{black} Because}
  \EQNN{x^\top \gamma_jx&=2{\rm tr}\bigg(\bigg(\sum_{i=1}^d\Phi_{j,i}x_i\bigg)\tilde{K}_j\bigg(\sum_{i'=1}^d\Phi_{j,i'}x_{i'}\bigg)\tilde{K}_j\bigg) \\
   &=2{\rm tr}\bigg(\tilde{K}_j^{1/2}\bigg(\sum_{i=1}^d\Phi_{j,i}x_i\bigg)\tilde{K}_j\bigg(\sum_{i'=1}^d\Phi_{j,i'}x_{i'}\bigg)\tilde{K}_j^{1/2}\bigg)\geq 0
  }
for $x=(x_1,\cdots,x_d)\in\mathbb{R}^d$, $\gamma_j$ is symmetric and nonnegative definite.
Hence $\Gamma$ is also symmetric and nonnegative definite almost surely under (B1)--(B3) and {\color{black} the fact} that
$\epsilon_n^2\sum_{j=1}^{m_n}\gamma_j({\color{black} \bar{X}_{j-1}^{n,\theta_0}})\overset{d}\to \Gamma$ as $n\to\infty$.
\end{remark}
}

\begin{theorem}\label{Malliavin-LAMN-thm}
Assume {\color{black}}% {\color[rgb]{0.8,0.3,0} (B1)--(B5)}. Then $\{P_{\theta,n}\}_{\theta,n}$ satisfies {\color{black} (L)}
 {\color{black}}% {\color[rgb]{0.8,0.3,0}  with {\color{black} $T(\theta_0)$ equal to} $\Gamma$ in (B4), $r_n=\epsilon_nI_d$,
$\dot{\xi}_j(\theta)$ and $\eta_j$ are defined by (\ref{N1-xi-def}) if (N1) is satisfied, and
\begin{equation*}
\dot{\xi}_j(\theta)=\frac{\partial_\theta p_j}{2\sqrt{p_j}}1_{\{p_j\neq 0\}}(x_j,\theta),
\quad \eta_j=\frac{\partial_\theta p_j}{2p_j}1_{\{p_j\neq 0\}}(x_j,\theta_0)
\end{equation*}
if (N2) is satisfied.} {\color{black} If further ($P'$) is satisfied, {\color{black} then} $\{P_{\theta,n}\}_{\theta,n}$ satisfies the LAMN property at $\theta=\theta_0$.}
\end{theorem}


{\color{black}}% {\color[rgb]{0.8,0.3,0} {\color{black} Note} that Theorem~\ref{Malliavin-LAMN-thm} works without {\color{black} having to identify} zero points of the density function $p_j$, unlike {\color{black} in} previous studies.
This is a {\color{black} major} advantage because it is often not {\color{black} an} easy task to show {\color{black} either} that $p_j$ has no zero points {\color{black} or} that zero points are common for every $\theta$.
For example, {\color{black} to our knowledge,} there are no results related to zero points of transition density functions
for the statistical model of multi-dimensional integrated diffusion processes.
In the following section, we see that Theorem~\ref{Malliavin-LAMN-thm} can be applied {\color{black} to} this model.}

{\color{black}
The following lemma is useful when we check (\ref{B4-stable-conv}) by using a martingale central limit theorem.
\begin{as}
The proof is {\color{black} given in} Section~\ref{Mcal-proof-section} {\color{black} of} the supplementary {\color{black} material}~\cite{fuk-ogi20s}.
\end{as}
\begin{arXiv}
{\color{black} The proof is left to Section~\ref{Mcal-proof-section} in the appendix.}
\end{arXiv}

{\color{black} For a matrix $A$, $\lVert A\rVert_{{\rm op}}$ denotes the operator norm of $A$.}
\begin{lemma}\label{B4-suff-lemma}
Assume (B1)--(B3) and that (\ref{Gj-def-eq}) is satisfied for any $A\in \mathcal{B}(\mathbb{R}^{k_j-k_{j-1}}\times \mathbb{R}^d)$. Then,
\begin{enumerate}
\item $E[G_j^n(G_j^n)^\top|\mathcal{G}_{j-1}]=\gamma_j({\color{black} \bar{X}_{j-1}^{n,\theta_0}})$ for any $1\leq j\leq m_n$, {\color{black} and}
\item $\epsilon_n^4\sum_{j=1}^{m_n}E[|G_j^n|^4|\mathcal{G}_{j-1}]\overset{P}\to 0$ as $n\to\infty$
if 
\begin{equation}
\epsilon_n^4m_n\alpha_n^8\bar{k}_n^8\sup_{j,i,\tilde{x}_{j-1}}
\lVert B_{j,i,\theta_0} \tilde{K}_j(\theta_0)+\tilde{K}_j(\theta_0)B_{j,i,\theta_0}^\top \rVert_{{\rm op}}^4\to 0.
\end{equation}
\end{enumerate}
\end{lemma}
}



{\color{black}
\subsection{The LAMN property for degenerate diffusion models}\label{degenerate-diffusion-subsection}

{\color{black} In this section, we show the LAMN property for degenerate diffusion processes
by applying the results in Sections~\ref{L2regu-subsection} and~\ref{LAMN-Malliavin-subsection}.}

Let $r\in\mathbb{N}$, and let $(\Omega,\mathcal{F},P)$ be the canonical probability space associated with {\color{black} an} $r$-dimensional Wiener process $W=\{W_t\}_{t\in [0,1]}$, that is,
$\Omega=C([0,1];\mathbb{R}^r)$, $P$ is the $r$-dimensional Wiener measure, $W_t(\omega)=\omega(t)$ for $\omega\in \Omega$, and $\mathcal{F}$ is the completion of {\color{black} the} Borel $\sigma$-field of $\Omega$ with respect to $P$.
Let $D$ be the {\color{black} Malliavin--Shigekawa} derivative related to the underlying Hilbert space $H=L^2([0,1];\mathbb{R}^r)$.
Let $\Theta$ be a bounded open convex set in $\mathbb{R}^d$.
We regard $\partial_xv=(\partial_{x_i}v_j)_{i,j}$ as a matrix for vectors $x$ and $v$.
$\bar{A}$ denotes the closure for a set $A$.

For $\theta\in \Theta$, let $X^\theta=(X_t^\theta)_{t\in [0,1]}$ be an $m$-dimensional diffusion {\color{black} process} satisfying $X_0^\theta=z_{{\rm ini}}$, and
\begin{equation}\label{degenerate-sde}
dX_t^\theta=b(X_t^\theta,\theta)dt+a(X_t^\theta,\theta)dW_t, \quad t\in [0,1],
\end{equation}
where $z_{{\rm ini}}\in \mathbb{R}^m$ and $a$ and $b$ are Borel functions.
We consider a statistical model with observations $(X_{j/n}^\theta)_{j=0}^n$.

Let $\kappa={\rm rank}(a(z,\theta))$.
Assume that $m/2\leq \kappa<m$ and {\color{black} that} $\kappa$ does not depend on $(z,\theta)$.
Then, by singular value decomposition, we can find an orthogonal matrix $U_{z,\theta}$ such that 
$U_{z,\theta}a(z,\theta)=(\tilde{a}(U_{z,\theta}z,\theta)^\top, O_{m-\kappa,r}^\top)^\top$,
where $O_{k,l}$ denotes {\color{black} a} $k\times l$ matrix with each element equal to zero.

{\color{black} First, we} assume that $U_{z,\theta}$ does not depend on $(z,\theta)$.
\begin{description}
\item[Assumption (C1).] The derivatives $\partial_z^i\partial_\theta^ja(z,\theta)$ and $\partial_z^i\partial_\theta^jb(z,\theta)$ exist on 
$\mathbb{R}^m\times \Theta$ and can be extended to continuous functions on 
$\mathbb{R}^m\times \bar{\Theta}$
for $i\in\mathbb{Z}_+$ and $0\leq j\leq 3$.
Moreover, $\sup_{z,\theta}(|\partial_zb(z,\theta)|\vee |\partial_za(z,\theta)|)<\infty$, 
and there exist an orthogonal matrix $U$ and $\mathbb{R}^\kappa\otimes \mathbb{R}^r$-, $\mathbb{R}^\kappa$-, and $\mathbb{R}^{m-\kappa}$-valued 
Borel functions $\tilde{a}(z,\theta)$, $\tilde{b}(z,\theta)$, and $\check{b}(z)$, respectively, such that 
\begin{equation}
Ua(z,\theta)=
\left(\begin{array}{c} \tilde{a}(Uz,\theta) \\ O_{m-\kappa,r} \end{array}\right), 
\quad Ub(z,\theta)=
\left(\begin{array}{c} \tilde{b}(Uz,\theta) \\ \check{b}(Uz) \end{array}\right)
\end{equation}
for any 
$z\in\mathbb{R}^m$ 
and $\theta\in{\color{black} \bar{\Theta}}$. Further, $\tilde{a}\tilde{a}^\top(z,\theta)$ is positive definite for any 
$(z,\theta)\in\mathbb{R}^m\times \bar{\Theta}$.
\end{description}
\begin{discuss}
{\color{black}}% {{\color[rgb]{0.8,0,0} 一階微分の有界性は解の存在だけならsupにthetaを入れる必要はないが，$DF$の一様評価などでthetaに関する一様性が必要}
\end{discuss}

There exists a unique strong solution $(X_t^\theta)_{t\in[0,1]}$ of (\ref{degenerate-sde}) under (C1).
Let $P_{\theta,n}$ be the distribution of $(X_{k/n}^\theta)_{k=0}^n$, and let $\theta_0$ be the true value of $\theta$.
We denote $X_t=X_t^{\theta_0}$.

Under (C1), by setting $Y_t^\theta=UX_t^\theta$, we obtain
\begin{equation}\label{Y-SDE}
dY_t^\theta=
\left(\begin{array}{c} \tilde{b}(Y_t^\theta,\theta) \\ \check{b}(Y_t^\theta) \end{array}\right)dt
+\left(\begin{array}{c} \tilde{a}(Y_t^\theta,\theta) \\ O_{m-\kappa,r} \end{array}\right)dW_t. 
\end{equation}

We denote $z=(x,y)$ for $x\in\mathbb{R}^\kappa$ and $y\in\mathbb{R}^{m-\kappa}$,
{\color{black} $\nabla_1=(\partial_{z_1},\cdots, \partial_{z_\kappa})$,
$\nabla_2=(\partial_{z_{\kappa+1}},\cdots, \partial_{z_m})$,
}
${\rm Ker}(A)=\{x\in \mathbb{R}^l;Ax=0\}$, and by $A^+$ the Moore--Penrose inverse for a $k\times l$ matrix $A$. 

\begin{description}
\item[Assumption (C2).] The derivative $\partial_z^i\check{b}(z)$ is bounded for $i\in\mathbb{N}$ and 
\EQQ{\sup_{z\in\mathbb{R}^m}\lVert(({\color{black} \nabla_1}\check{b})^\top {\color{black} \nabla_1}\check{b})^{-1}(z)\rVert_{{\rm op}}<\infty.}
Moreover, 
  \EQN{{\rm Ker}(\tilde{a}(z,\theta))&\subset {\rm Ker}(\partial_{\theta_i}\tilde{a}(z,\theta)), \\
   {\rm Ker}(({\color{black} \nabla_1}\check{b})^\top(z))&\subset {\rm Ker}(({\color{black} \nabla_1}\check{b})^\top(z)\partial_{\theta_i}\tilde{a}\tilde{a}^+(z',\theta))
  }
for any $z,z'\in\mathbb{R}^m$, $1\leq i\leq d$, and $\theta \in\Theta$.
Furthermore, at least one of the following two conditions is satisfied;
\begin{enumerate}
\item $\check{b}$ is bounded;
\item ${\color{black} \nabla_2}\tilde{a}(z,\theta)=0$ and ${\color{black} \nabla_2}\check{b}(z)=0$ for any {\color{black} $z\in\mathbb{R}^m$} and $\theta\in\Theta$.
\end{enumerate} 
\end{description}
We need (C2) to satisfy $\partial_\theta F_{n,\theta,j}=B_{j,i,\theta}F_{n,\theta,j}$ in (B3). 
\begin{as}
See Section~\ref{degenerate-lamn-section} {\color{black} of the supplementary material~\cite{fuk-ogi20s} for the details}.
\end{as}
\begin{arXiv}
See Section~\ref{degenerate-lamn-section} in the appendix for the details.
\end{arXiv}
\begin{discuss}
{\color{black}}% {{\color[rgb]{0.8,0,0} 局所化しないなら$z'=z$でもいい}
\end{discuss}

{\color{black} We can write $\tilde{a}^+=\tilde{a}^\top(\tilde{a}\tilde{a}^\top)^{-1}$ because $\tilde{a}\tilde{a}^\top$ is invertible.}
If $r=\kappa$ and (C1) is satisfied, {\color{black} then} we can easily check ${\rm Ker}(\tilde{a}(z,\theta))\subset {\rm Ker}(\partial_{\theta_i}\tilde{a}(z,\theta))$
{\color{black} because} $\tilde{a}(z,\theta)$ is invertible. Similarly, we can easily check 
${\rm Ker}(({\color{black} \nabla_1}\check{b})^\top(z))\subset {\rm Ker}(({\color{black} \nabla_1}\check{b})^\top(z)\partial_{\theta_i}\tilde{a}\tilde{a}^+(z',\theta))$
if $m-\kappa=\kappa$ and $({\color{black} \nabla_1}\check{b})^\top {\color{black} \nabla_1}\check{b}(z)$ is positive definite.

Let $\Psi_{t,\theta}=({\color{black} \nabla_1}\check{b})^\top\tilde{a}\tilde{a}^\top{\color{black} \nabla_1}\check{b}(UX_t,\theta)$, and let
  \EQN{\label{Gamma-def}
   \Gamma&=\bigg(\frac{1}{2}\int^1_0{\rm tr}((aa^\top)^+\partial_{\theta_i}(aa^\top)(aa^\top)^+\partial_{\theta_j}(aa^\top))(X_t,\theta_0)dt \\
   & \quad \quad +\frac{1}{2}\int^1_0{\rm tr}(\Psi_{t,\theta_0}^{-1}\partial_{\theta_i}\Psi_{t,\theta_0}\Psi_{t,\theta_0}^{-1}\partial_{\theta_j}\Psi_{t,\theta_0})dt\bigg)_{1\leq i,j\leq d}.
  }

\begin{description}
\item[Assumption (C3).] $\Gamma$ is positive definite almost surely.
\end{description}

\begin{theorem}\label{degenerate-LAMN-thm}
Assume (C1)--(C3). Then $\{P_{\theta,n}\}_{\theta,n}$ satisfies the LAMN property at $\theta=\theta_0$ with $\Gamma$.
\end{theorem}


\begin{remark}
The proof of Theorem~\ref{degenerate-LAMN-thm} in Section~\ref{degenerate-lamn-section} 
\begin{as}
{\color{black} of the supplementary file~\cite{fuk-ogi20s}}
\end{as}
shows that we obtain similar results when $\kappa=m$ and $aa^\top$ is positive definite
by ignoring $\check{b}$ and $\Psi_{t,\theta}$. This approach {\color{black} allows} another proof of the LAMN property for nondegenerate diffusion processes by Gobet~\cite{gob01}.
\end{remark}

\begin{remark}
The first term in the right-hand side of (\ref{Gamma-def}) is equal to $\Gamma$ in Gobet~\cite{gob01} for nondegenerate diffusion processes.
It is also the same as $\Gamma$ for the statistical model with observations $\{[Y_{j/n}]_l\}_{0\leq j\leq n, 1\leq l\leq \kappa}$.
Then, the second term in the right-hand side of (\ref{Gamma-def}) corresponds to additional information obtained by 
observation $\{[Y_{j/n}]_l\}_{0\leq j\leq n, \kappa+1\leq l\leq m}$ for the degenerate process.
\end{remark}



\begin{example}\label{degenerate-ex1}
Let $\kappa\in\mathbb{N}$. Let $X_t^\theta$ and $\bar{X}_t^\theta$ be {\color{black} a $\kappa$-dimensional diffusion process} satisfying
\begin{equation}
dX_t^\theta=d(X_t^\theta,\bar{X}_t^\theta,\theta)dt+c(X_t^\theta,\theta)dW_t,
\quad d\bar{X}_t^\theta=X_t^\theta dt, \quad t\in [0,1],
\end{equation}
where $\theta \in \Theta\subset \mathbb{R}^d$ and $W_t$ is a $\kappa$-dimensional standard Wiener process.
We assume that $cc^\top(x,\theta)$ is positive definite and $c(x,\theta)$ and $d(z,\theta)$ are smooth functions with bounded derivatives $\partial_xc$ and $\partial_zd$.
Then, (C1) and (C2) are satisfied with $U=I_{2\kappa}$. $\Gamma$ is given by
\begin{equation}\label{Gamma-ex}
\Gamma=\left(\int^1_0{\rm tr}((cc^\top)^{-1}\partial_{\theta_i}(cc^\top)(cc^\top)^{-1}\partial_{\theta_j}(cc^\top))(X_t^{\theta_0},\theta_0)dt\right)_{1\leq i,j\leq d}.
\end{equation}
If further $\Gamma$ is positive definite almost surely, {\color{black} then} we obtain the LAMN property of this model by Theorem~\ref{degenerate-LAMN-thm}. 
\end{example}
\begin{discuss}
{\color{black}}% {{\color[rgb]{0.8,0,0} $\Gamma$の非退化性を言うにはGloter Jacod(2001b)のidentifiability assumptionをチェックすればよいが，$\Theta$が多次元の時は
この$\Gamma$の形からいうのと変わらないか．Uchida Yoshidaのnondegeneracyは退化しているSDEでは多分使えない}
\end{discuss}

{\color{black} Example~\ref{degenerate-ex1} {\color{black} pertains to Langevin-type} molecular dynamics (\ref{Langevin-eq}).
Here we assumed that the position $X_t^\theta$ and {\color{black} velocity} $\bar{X}_t^\theta$ of {\color{black} a} molecule are observed at discrete time points.
In {\color{black} Example~\ref{int-obs-ex} of} Section~\ref{integrated-diffusion-subsection}, we deal with the case where we {\color{black} observe only} {\color{black} the position} $\bar{X}_t^\theta$.

With some restriction on the diffusion coefficient $c$, we can extend Example~\ref{degenerate-ex1} 
to the case {\color{black} where} ${\rm dim}(X_t^\theta)>{\rm dim}(\bar{X}_t^\theta)$.
\begin{example}
Let $\kappa'\leq \kappa$. Let $X_t^\theta$ be the same as in Example~2.1, 
and let $\bar{X}_t^\theta$ be a $\kappa'$-dimensional stochastic process satisfying
$[\bar{X}_t^\theta]_i=\int^t_0[X_s^\theta]_ids$ for $1\leq i\leq \kappa'$.
Moreover, let $c(x,\theta)=f(x,\theta)A$ for some $\mathbb{R}$-valued function $f$ and matrix $A$ independent of $x$ and $\theta$.
We assume that $AA^\top$ is positive definite, $f$ is positive-valued, 
and $f(x,\theta)$ and $d(z,\theta)$ are smooth functions
with bounded derivatives $\partial_xf$ and $\partial_zd$.
Then, (C1) and (C2) are satisfied {\color{black} because} $({\color{black} \nabla_1}\check{b})^\top=(I_{\kappa'} \ O_{\kappa',\kappa -\kappa'})$
and $\partial_\theta cc^{-1}(x,\theta)=\partial_\theta f f^{-1}(x,\theta)I_\kappa$.
We have $\Psi_{t,\theta}=f^2(X_t^\theta,\theta)([AA^\top]_{ij})_{1\leq i,j\leq \kappa'}$, and hence we have
  \EQNN{[\Gamma]_{ij}&=\frac{1}{2}\int^1_0\bigg\{\frac{2\partial_{\theta_i}f}{f}\frac{2\partial_{\theta_j}f}{f}(X_t,\theta_0)\cdot \kappa
   +\frac{2\partial_{\theta_i}f}{f}\frac{2\partial_{\theta_j}f}{f}(X_t,\theta_0)\cdot \kappa'\bigg\}dt \\
   &=2(\kappa+\kappa')\int^1_0\frac{\partial_{\theta_i}f\partial_{\theta_j}f}{f^2}(X_t,\theta_0)dt.
  }

If we only observe $(X_{k/n}^\theta)_{k=0}^n$, then $\Gamma$ in Gobet~\cite{gob01} is calculated as
\begin{equation*}
\Gamma=\bigg(2\kappa\int^1_0\frac{\partial_{\theta_i}f\partial_{\theta_j}f}{f^2}(X_t,\theta_0)dt\bigg)_{1\leq i,j\leq d}.
\end{equation*}
Therefore, we conclude that $\Gamma$ for observations $X_t^\theta$ and $\bar{X}_t^\theta$ 
is $(\kappa+\kappa')/\kappa$ times as much as the one for observations $X_t^\theta$.
\end{example}

}


\begin{example}
Let $X_t^\theta=(X_t^{\theta,1},X_t^{\theta,2})$ be a two-dimensional diffusion process satisfying
\begin{equation}
\left\{
\begin{array}{ll}
dX_t^{\theta,1}=&(d(X_t^{\theta,1},X_t^{\theta,2},\theta)+e(X_t^{\theta,1},X_t^{\theta,2}))dt+c(X_t^{\theta,1},X_t^{\theta,2},\theta)dW_t, \\
dX_t^{\theta,2}=&d(X_t^{\theta,1},X_t^{\theta,2},\theta)dt+c(X_t^{\theta,1},X_t^{\theta,2},\theta)dW_t, \\
\end{array}
\right.
\end{equation}
where $\theta \in \Theta\subset \mathbb{R}^d$ and $W_t$ is a one-dimensional standard Wiener process.
That is, {\color{black} the} diffusion coefficients of $X_t^{\theta,1}$ and $X_t^{\theta,2}$ are the same.
We assume that $c$ is positive-valued, $\sup_{x,y}|\partial_xe(x,y)|^{-1}<\infty$, and $c(x,y,\theta)$, $d(x,y,\theta)$, and $e(x,y)$ are smooth functions 
with bounded derivatives $\partial_zc$, $\partial_zd$, and $\partial_z^ie$ for $i\in\mathbb{N}$ (z=(x,y)). 
Moreover, we assume that at least one of the following two conditions holds true:
\begin{enumerate}
\item $e$ is bounded;
\item $e(x,y)=\tilde{e}(x+y)$ and $c(x,y,\theta)=\tilde{c}(x+y,\theta)$ for some functions $\tilde{e}$ and $\tilde{c}$.
\end{enumerate}
Then (C1) and (C2) are satisfied with
\begin{equation*}
U=\frac{1}{\sqrt{2}}\left(\begin{array}{cc}
1 & 1 \\
1 & -1
\end{array}\right),
\end{equation*}
{\color{black} and} $\Gamma$ is given by (\ref{Gamma-ex}).
If further $\Gamma$ is positive definite almost surely, {\color{black} then} we obtain the LAMN property of this model by Theorem~\ref{degenerate-LAMN-thm}. 

For the statistical model with observations $(X_{j/n}^{\theta,2})_{j=0}^n$, $\Gamma$ is equal to half of the one in (\ref{Gamma-ex}).
The above result shows that the efficient asymptotic variance for estimators does not depend on $e$ and is equal to just half of the one when we observe $(X_{j/n}^{\theta,2})_{j=0}^n$.
\end{example}
}

{\color{black}
\begin{example}
Let $m/2\leq \kappa<m$. Let $X_t^\theta$ be an $m$-dimensional diffusion process satisfying
\begin{equation}\label{factor-model}
dX_t^\theta=e(X_t^\theta)dt+f(X_t^\theta,\theta)AdW_t,
\end{equation}
where $W$ is a $\kappa$-dimensional standard Wiener process, $f(z,\theta)$ is an $\mathbb{R}$-valued function,
and $A$ is an $m\times\kappa$ matrix independent of $z$ and $\theta$.
Let $U^\top(\Lambda \ O_{\kappa, m-\kappa})^\top V$ be {\color{black} the} singular value decomposition of $A$ for a $\kappa\times \kappa$ diagonal matrix $\Lambda$,
and orthogonal matrices $U$ and $V$ of size $m$ and $\kappa$, respectively.
Then we have 
\begin{equation*}
Uf(z,\theta)A=\left(
\begin{array}{c}
\tilde{c}(Uz,\theta) \\
O_{m-\kappa,\kappa}
\end{array}
\right), \quad Ue(z)=\left(
\begin{array}{c}
\tilde{e}(Uz) \\
\check{e}(Uz)
\end{array}
\right),
\end{equation*}
where $\tilde{c}(z,\theta)=f(U^\top z,\theta)\Lambda V$, and $\tilde{e}(z)$ and $\check{e}(z)$ are suitable functions. 

We assume that ${\rm rank}(A)=\kappa$ (that is, $\Lambda$ is invertible), $f$ is positive--valued, 
$f(z,\theta)$ and $e(z)$ are smooth functions, 
and $\partial_zf$, $\partial_z^ie$, and $\lVert (({\color{black} \nabla_1}\check{e})^\top {\color{black} \nabla_1}\check{e})^{-1}\rVert_{{\rm op}}$ are bounded for $i\in\mathbb{Z}_+$.
Then we obtain $\partial_\theta \tilde{c}\tilde{c}^{-1}(z,\theta)=\partial_\theta ff^{-1}(U^\top z,\theta)I_\kappa$,
and consequently (C1) and (C2) hold.
Moreover, we have 
  \EQQ{\Psi_{t,\theta}=f^2(X_t^{\theta_0},\theta)(({\color{black} \nabla_1}\check{e})^\top \Lambda^2 {\color{black} \nabla_1} \check{e})(UX_t^{\theta_0})}
and hence
\begin{equation*}
\Gamma=\bigg(2m\int^1_0\frac{\partial_{\theta_i}f\partial_{\theta_j}f}{f^2}(X_t^{\theta_0},\theta_0)dt\bigg)_{1\leq i,j\leq d}.
\end{equation*}
If $\Gamma$ is positive definite almost surely, {\color{black} then} we have the LAMN property of this model.

We can regard (\ref{factor-model}) as a multi-factor model for stock prices, where each component of $W$ is regarded as a factor 
{\color{black} that influences the} stock prices, $A$ {\color{black} comprises the} contributions of each factor to each stock, and $f$ is a scalar {\color{black} that depends} on {\color{black} the} stock prices.
The above results show that we obtain the LAMN property of such a degenerate model if the number $\kappa$ of factors is in $[m/2,m)$.
\end{example}

\begin{discuss}
{\color{black}}% {{\color[rgb]{0.8,0,0}
\begin{equation*}
dX_t=e(X_t)dt+Ac(X_t,\theta)dW_t
\end{equation*}
で$A$: $m\times \kappa$行列，$c$: $\kappa\times \kappa$行列，$A=U^\top \Lambda$, $c=E(\theta)C(X_t)$, $E={\rm diag}((\theta_i)_{1\leq i\leq \kappa})$とすると，
$\partial_{\theta_i}\tilde{c}\tilde{c}^+=E_i\theta_i^{-1}$と書け，
${\rm Ker}(({\color{black} \nabla_1}\check{e})^\top)$が$(1,0,\cdots, 0), (0,1,0,\cdots, 0), \cdots, (0,\cdots, 0,1)$のいくつかで張られていたらOKだが，
これを満たす例がわかりづらいか 
}
\end{discuss}


\subsection{The LAMN property for partial observations}\label{integrated-diffusion-subsection}

In this section, we show the LAMN property for degenerate diffusion processes with partial observations.
Gloter and Gobet~\cite{glo-gob08} showed the LAMN property for a one-dimensional integrated diffusion process.
{\color{black} While this model is similar to the one in Example~\ref{degenerate-ex1}, 
{\color{black} the} observations are only the integrated process $\bar{X}_t^{\theta_0}$ (partial observations).}
Their proof depends on Aronson's estimate, which is difficult to obtain for the multi-dimensional process.
We can avoid {\color{black} the} Aronson-type estimate by using the {\color{black} scheme with the $L^2$ regularity condition} 
in Sections \ref{L2regu-subsection} and \ref{LAMN-Malliavin-subsection},
and {\color{black} can} consequently extend their results to {\color{black} the} multi-dimensional case.
We can also generalize {\color{black} the} observed components of $X_t^{\theta_0}$ and $\bar{X}_t^{\theta_0}$, which yields 
an interesting example of a stock process and integrated volatility observations (Example~\ref{int-vol-ex}).

Let $m\in\mathbb{N}$, and let $(\Omega,\mathcal{F},P)$, $W$, $D$, $H$, and $\Theta$ be the same as {\color{black} in} Section~\ref{degenerate-diffusion-subsection}.
{\color{black} We consider a process} $Y_t^\theta=(\tilde{Y}_t^\theta,\check{Y}_t^\theta)$
{\color{black} {\color{black} that} satisfies a slight restricted version of the stochastic differential equation (\ref{Y-SDE}): 
$(\tilde{Y}_0^\theta,\check{Y}_0^\theta)=(\tilde{z}_{{\rm ini}},\check{z}_{{\rm ini}})$, and}
\EQN{d\tilde{Y}_t^\theta&=\tilde{b}(\tilde{Y}_t^\theta,\check{Y}_t^\theta,\theta)dt+\tilde{a}(\tilde{Y}_t^\theta,\theta)dW_t, \\
d\check{Y}_t^\theta&=B\tilde{Y}_t^\theta dt,}
{\color{black} where $B$ is an $(m-\kappa)\times \kappa$ matrix such that $BB^\top$ is positive definite.
Let ${\color{black} \mathcal{Q}}:\mathbb{R}^\kappa\to \mathbb{R}^\kappa$ {\color{black} be a projection}.
We assume that $({\color{black} \mathcal{Q}}\tilde{Y}_{k/n}^{\theta_0})_{k=0}^n$ and $({\color{black} \check{Y}_{k/n}^{\theta_0}})_{k=0}^n$ are observed.
Let {\color{black} $q_1={\rm rank}(\mathcal{Q})$, $q_2=m-\kappa$}, and let $q=q_1+q_2$. We assume that {\color{black} $0\leq q_1< \kappa$}.

For a $k\times l$ matrix $A$, we denote ${\rm Im}(A)=\{Ax;x\in\mathbb{R}^l\}$. }
\begin{description}
\item[Assumption (C2$'$).] 
The derivatives $\partial_x^i\partial_\theta^j\tilde{a}(x,\theta)$ and $\partial_z^i\partial_\theta^j\tilde{b}(z,\theta)$ exist on $\mathbb{R}^m\times \Theta$ 
and can be extended to continuous functions on $\mathbb{R}^m\times \bar{\Theta}$ for $i\in\mathbb{Z}_+$ and $0\leq j\leq 3$.
Moreover, $\sup_{z,\theta}(|\partial_z\tilde{b}(z,\theta)|\vee |\partial_x\tilde{a}(x,\theta)|)<\infty$,
${\rm Ker}(\tilde{a}(x,\theta))\subset {\rm Ker}(\partial_{\theta_i}\tilde{a}(x,\theta))$,
and ${\rm Ker}(B)\subset {\rm Ker}(B\partial_{\theta_i}\tilde{a}\tilde{a}^+(x,\theta))$
for any $x\in\mathbb{R}^\kappa$, $1\leq i\leq d$, and $\theta \in\Theta$.
\item[Assumption (C4).] 
\EQ{\label{Q1Q2-cond}  {\rm Ker}({\color{black} B})\subset {\rm Im}({\color{black} \mathcal{Q}}),}
{\color{black} and}
\begin{equation}\label{Q1Q2-cond2}
{\color{black} \mathcal{Q}}\partial_{\theta_i} \tilde{a}\tilde{a}^+(x,\theta)=\partial_{\theta_i} \tilde{a}\tilde{a}^+(x,\theta){\color{black} \mathcal{Q}}
\end{equation}
for any $1\leq i\leq d$, $x\in\mathbb{R}^\kappa$, and $\theta\in\Theta$.
\end{description}
\begin{discuss}
{\color{black}}% {{\color[rgb]{0.8,0,0} $(I-Q_2)y$に依存させることもできるが，$\tilde{b}$が$(I-Q_2)y$に依存するかどうかで
${\rm Im}(B{\color{black} \mathcal{Q}})\supset ({\rm Im}(Q_2))^\perp$を仮定したり，$F$や$K$の次元が変わったりするからやめておく．}
\end{discuss}
{\color{black} 
By (\ref{Q1Q2-cond}), we have
  \EQ{q_1={\color{black} \dim {\rm Im}}({\color{black} \mathcal{Q}})\geq \dim {\rm Ker}({\color{black} B})=\kappa-\dim {\rm Im}({\color{black} B})= \kappa-q_2,}
which implies $q\geq \kappa$.

Let {\color{black} $R_1:{\rm Im}(\mathcal{Q})\to \mathbb{R}^{q_1}$} and $R_3:{\rm Im}(I_\kappa-{\color{black} \mathcal{Q}})\to \mathbb{R}^{\kappa-q_1}$
be any isomorphism on vector spaces. We denote $\tilde{Q}_1=R_1{\color{black} \mathcal{Q}}$, $\tilde{Q}_2={\color{black} B}$, and $\tilde{Q}_3=R_3(I_\kappa-{\color{black} \mathcal{Q}})$.
For a $\kappa\times \kappa$ matrix $A$, we denote}
$\Upsilon_{i,j}(A)=\tilde{Q}_iA\tilde{Q}_j^\top$ for $1\leq i,j\leq 3$,
\EQQ{\Xi_1(A)=\MAT{cc}{\Upsilon_{1,1} & \Upsilon_{1,2}/2 \\ \Upsilon_{2,1}/2 & \Upsilon_{2,2}/3}(A),
\quad \Xi_2(A)=\MAT{cc}{O_{q_1,q_1} & \Upsilon_{1,2}/2 \\ O_{q_2,q_1} & \Upsilon_{2,2}/6}(A),}
\EQQ{\Xi_3(A)=\MAT{cc}{\Upsilon_{1,1} & \Upsilon_{1,2}/2 \\ \Upsilon_{2,1}/2 & 2\Upsilon_{2,2}/3}(A),}
{\color{black} and for $L\in\mathbb{N}$, $L\geq 3$ and $1\leq k,l\leq 2$, 
we define an $(Lq+{\color{black} (\kappa-q_1)}(k-1))\times (Lq+{\color{black} (\kappa-q_1)}(l-1))$ matrix $\psi_L^{k,l}(A)$ by}
  \EQQ{\psi_L^{1,1}(A)=\MAT{ccccc}{
   \Xi_1 & \Xi_2 & O_{q,q} & \cdots & O_{q,q} \\
   \Xi_2^\top & \Xi_3 & \ddots & \ddots & \vdots \\
   O_{q,q} & \ddots & \ddots & \ddots & O_{q,q} \\
   \vdots & \ddots & \ddots & \Xi_3 & \Xi_2 \\
   O_{q,q} & \cdots & O_{q,q} & \Xi_2^\top & {\color{black} \Xi_3}
   }(A), 
  }
  \EQQ{
   \psi_L^{1,2}(A)=\MAT{cc}{
    & O_{(L-1)q,\kappa-q_1} \\
   \psi_L^{1,1} & \Upsilon_{1,3}/2 \\
    & \Upsilon_{2,3}/6
   }(A), \quad \psi_L^{2,1}(A)=(\psi_L^{1,2})^\top(A),
  }
  \EQQ{
   \psi_L^{2,2}(A)=\MAT{cccc}{
   \multicolumn{4}{c}{\psi_L^{1,2}} \\ 
   {\color{black} O_{\kappa-q_1,(L-1)q}} & \Upsilon_{3,1}/2 & \Upsilon_{3,2}/6 & \Upsilon_{3,3}/3
   }(A).
  }
{\color{black} Here we ignore $\Upsilon_{i,j}$ if ${\rm rank}(\tilde{Q}_i)=0$ or ${\rm rank}(\tilde{Q}_j)=0$. Let}
  \EQNN{\mathcal{T}_{k,l,L}({\color{black} x})&=\Big({\rm tr}(\partial_{\theta_i}(\psi_L^{k,k}(\tilde{a}\tilde{a}^\top)^{-1})({\color{black} x},\theta_0)
   \psi_L^{k,l}(\tilde{a}\tilde{a}^\top)({\color{black} x},\theta_0) \\
   &\quad \quad \times \partial_{\theta_j}(\psi_L^{l,l}(\tilde{a}\tilde{a}^\top)^{-1})({\color{black} x},\theta_0)
   \psi_L^{l,k}(\tilde{a}\tilde{a}^\top)({\color{black} x},\theta_0))\Big)_{1\leq i,j\leq d}.
  }

{\color{black} To show the LAMN property of partial observations, we consider an augmented model generated by block observations
with some observations of $(I_\kappa-\mathcal{Q})\tilde{Y}$, following the idea of Gloter and Gobet~\cite{glo-gob08}.
The matrix $\psi^{k,l}_L$ corresponds to {\color{black} the} covariance matrix of {\color{black} the} block observations.
See Sections~\ref{aug-L-subsection} and~\ref{ori-L-subsection} 
\begin{as}
{\color{black} of} the supplementary {\color{black} material}~\cite{fuk-ogi20s} 
\end{as}
for the details.}
\begin{description}
\item[Assumption (C5).] 
{\color{black} There} exists an $\mathbb{R}^d\otimes \mathbb{R}^d$-valued continuous function $g({\color{black} x})$ such that
\EQ{\label{C5-eq} {\color{black} L^{-1}\mathcal{T}_{k,l,L}(x)\to g(x)}}
{\color{black} as $L\to\infty$ uniformly in $x$ on compact sets} for $1\leq k,l\leq 2$.
\end{description}
\begin{discuss}
{\color{black}}% {{\color[rgb]{0.8,0,0} (C5)は局所化した後に$L^{-1}\mathcal{T}_{k,l,L}(x)\to g(x)$の一様収束が成り立ち，
$\sum_j\gamma_j\overset{P}\to \Gamma$, $\sup_nE[n^{-1}\sum_j|\gamma_j|]<\infty$等に使う．}
\end{discuss}

Let
\EQ{\label{Gamma'-def} \Gamma'={\color{black} \frac{1}{2}\int^1_0g(\tilde{Y}_t)dt}.}

\begin{description}
\item[Assumption (C6).] $\Gamma'$ is positive definite almost surely.
\end{description}

Let $P_{\theta,n}$ be the distribution of partial observations $({\color{black} \mathcal{Q}}\tilde{Y}_{k/n}^{\theta})_{k=0}^n$ and $({\color{black} \check{Y}_{k/n}^{\theta}})_{k=0}^n$.
\begin{theorem}\label{partial-LAMN-thm}
Assume (C2$'$) {\color{black} and} (C4)--(C6). Then $\{P_{\theta,n}\}_{\theta,n}$ satisfies the LAMN property at $\theta=\theta_0$ with $\Gamma'$.
\end{theorem}


\begin{example}[Integral observations]\label{int-obs-ex}
Let $X_t^\theta$ and $\bar{X}_t^\theta$ be the same as {\color{black} in} Example~\ref{degenerate-ex1}.
We consider a statistical model with {\color{black} observations $(\bar{X}_{k/n}^{\theta_0})_{k=0}^n$}.
In this case, we have $m=2\kappa$, $B=I_\kappa$, ${\color{black} \mathcal{Q}=O_{\kappa,\kappa}}$.
We assume that $cc^\top(x,\theta)$ is positive definite and {\color{black} that} $c(x,\theta)$ and $d(z,\theta)$ are smooth functions 
with bounded derivatives $\partial_xc$ and $\partial_zd$.
{\color{black} As in} Example~\ref{degenerate-ex1}, we have (C2$'$). Moreover, we can check (C4).
\begin{discuss}
{\color{black}}% {{\color[rgb]{0.8,0,0} $\Theta$を$\theta_0$の周りで取り直せば$\bar{\Theta}$上のsmoothへの拡張はOK.}
\end{discuss}

We can see that $\psi_L^{2,2}(A)=V_L\otimes A$,
where {\color{black} $\otimes$ denotes} the Kronecker {\color{black} product}, and $V_L$ is {\color{black} an} $(L+1)\times (L+1)$ matrix satisfying 
\EQ{\label{VL-def} [V_L]_{ij}=(2/3)1_{\{i=j\}}+(1/6)1_{\{|i-j|=1\}}-(1/3)1_{\{i=j \ {\rm and} \ i\in\{1,L+1\}\}}.}

{\color{black} {\color{black} Because} we obtain similar equations for $\psi_L^{1,1}(A)$ and $\psi_L^{1,2}(A)$,
\begin{as}
together with Lemma~\ref{block-mat-lem} {\color{black} in the supplementary {\color{black} material}~\cite{fuk-ogi20s}},
\end{as} 
\begin{arXiv}
together with Lemma~\ref{block-mat-lem},
\end{arXiv}
we have (\ref{C5-eq}) for
\EQ{\label{int-g-eq} g(x)=({\rm tr}((cc^\top)^{-1}\partial_{\theta_i}(cc^\top)(cc^\top)^{-1}\partial_{\theta_j}(cc^\top))(x,\theta_0))_{1\leq i,j\leq d}.}
\begin{discuss}
{\color{black}}% {{\color[rgb]{0.8,0,0} 適当な$V_L^{1,2}$に対し，$\psi_L^{1,2}(A)=V_L^{1,2}\otimes A$. よって混合積性質から(\ref{C5-eq})が成立．
${\rm tr}(A\otimes B)={\rm tr}(A){\rm tr}(B)$. $cc^\top$は$x$に関してsmoothだからC5の収束は広義一様収束．}
\end{discuss}
Therefore, we have the LAMN property of this model if $\Gamma'$ in (\ref{Gamma'-def}) is positive definite almost surely.

This result is an extension of Gloter and Gobet~\cite{glo-gob08} to multi-dimensional processes.
Moreover, the result can be applied to the Langevin-type molecular dynamics in (\ref{Langevin-eq}) with {\color{black} positional} observations.
}
\end{example}

\begin{remark}\label{integrated-rem}
If we observe $(X_{k/n}^{\theta_0})_{k=0}^n$ instead of $(\bar{X}_{k/n}^{\theta_0})_{k=0}^n$, then Gobet~\cite{gob01} shows the LAMN property for {\color{black} this} model
with $\Gamma$ the same as (\ref{Gamma'-def}) {\color{black} and (\ref{int-g-eq})}. On the other hand, if we observe both $(X_{k/n}^{\theta_0})_{k=0}^n$ and $(\bar{X}_{k/n}^{\theta_0})_{k=0}^n$,
{\color{black} then} Example~\ref{degenerate-ex1} shows the LAMN property with $\Gamma$ twice {\color{black} that} in (\ref{Gamma'-def}).
Therefore, we can say that the efficient asymptotic variance with observations $(X_{k/n}^{\theta_0})_{k=0}^n$ and $(\bar{X}_{k/n}^{\theta_0})_{k=0}^n$ 
is half of {\color{black} that} with observations $(X_{k/n}^{\theta_0})_{k=0}^n$ or {\color{black} half of that} with $(\bar{X}_{k/n}^{\theta_0})_{k=0}^n$.
\end{remark}


\begin{example}[Observations of {\color{black} a} stock process and {\color{black} integrated} volatility] \label{int-vol-ex}
{\color{black} Let $W$ be a} two-dimensional standard Wiener process, {\color{black} and let} 
$c$ {\color{black} be an} $\mathbb{R}^2\otimes \mathbb{R}^2$-valued function {\color{black} with $c^j$ for $j\in\{1,2\}$.
Let $X_t={\color{black} (X_t^i)_{i=1}^3}$ be a four-dimensional process satisfying}
\EQN{
dX_t^1&=d^1({\color{black} X_t},\theta)dt+c^1(X_t^1,X_t^2,\theta)dW_t, \\
dX_t^2&=d^2({\color{black} X_t},\theta)dt+c^2(X_t^1,X_t^2,\theta)dW_t, \\
d{\color{black} X_t^3}&=X_t^2dt.
}
{\color{black} We assume {\color{black} that we observe} $((X_{k/n}^1,{\color{black} X_{k/n}^3}))_{k=0}^n$. In this case, we have {\color{black} $m=3$}, $\kappa=2$, $r=2$, 
{\color{black} \EQ{{\color{black} \mathcal{Q}}=\MAT{cc}{1 & 0 \\ 0 & 0}, \quad B=(0 \ 1).}}
}

We assume that $cc^\top(x,\theta)$ is positive definite {\color{black} for each $(x,\theta)$,} and $c(x,\theta)$, $d^1(z,\theta)$, and $d^2(z,\theta)$ are smooth functions 
with bounded derivatives $\partial_xc$, $\partial_zd^1$, and $\partial_zd^2$.
{\color{black} We can check (\ref{Q1Q2-cond}).}

{\color{black} We consider the following two cases.
\begin{enumerate}
\item The case {\color{black} where} $c(x_1,x_2,\theta)=f(x_1,x_2,\theta)A$ for a matrix A and a positive--valued function $f(x_1,x_2,\theta)$:

We have that $AA^\top$ is positive definite and $\partial_{\theta_i}\tilde{a}\tilde{a}^+=\partial_{\theta_i}cc^{-1}=\partial_{\theta_i}ff^{-1}I_2$.
Then {\color{black} (C2$'$) and (C4) are satisfied}.
Moreover, we obtain 
  \EQ{\psi_L^{k,l}(\partial_{\theta_i}(\tilde{a}\tilde{a}^\top))=\frac{2\partial_{\theta_i}f}{f}\psi_L^{k,l}(\tilde{a}\tilde{a}^\top).}
\begin{as}
Together with Lemma~\ref{block-mat-lem} {\color{black} in the supplementary {\color{black} material}~\cite{fuk-ogi20s}},
\end{as} 
\begin{arXiv}
Together with Lemma~\ref{block-mat-lem}, 
\end{arXiv}
we have (C5) with 
  \EQQ{g(x_1,x_2)=\bigg(\frac{8\partial_{\theta_i}f\partial_{\theta_j}f}{f^2}(x_1,x_2,\theta_0)\bigg)_{i,j}.}
Therefore, we have the LAMN property if $\Gamma'$ in (\ref{Gamma'-def}) is positive definite almost surely.

\item The case {\color{black} where} $c(x_1,x_2,\theta)$ is a diagonal matrix for any $(x_1,x_2,\theta)$:\\
{\color{black} Because} $\partial_{\theta_i}\tilde{a}\tilde{a}^+$ also becomes a diagonal matrix, {\color{black} (C2$'$) and (C4) are satisfied}.
Moreover, we have $\Upsilon_{1,2}=0$ and $\Upsilon_{1,3}=0$. Then, by rearranging {\color{black} the} rows and columns of $\psi_L^{2,2}$
by using an orthogonal matrix $\mathcal{V}_L$ of size $2L+1$, we have
\EQ{\mathcal{V}_L\psi_L^{2,2}(M)\mathcal{V}_L^\top=\MAT{cc}{[M]_{11}I_L & O_{L,L} \\ O_{L,L} & [M]_{22}V_L}}
for any diagonal matrix $M$ of size $2$, where $V_L$ {\color{black} is} defined in (\ref{VL-def}).
Together with Lemma~\ref{block-mat-lem} 
\begin{as}
{\color{black} in the supplementary {\color{black} material}~\cite{fuk-ogi20s}}
\end{as}
and similar equations for $\psi_L^{1,2}$ and $\psi_L^{1,1}$, we have (\ref{C5-eq}) with
\EQ{g(x_1,x_2)=\bigg(\bigg(\frac{4\partial_{\theta_i}[c]_{11}\partial_{\theta_j}[c]_{11}}{[c]_{11}^2}
+\frac{4\partial_{\theta_i}[c]_{22}\partial_{\theta_j}[c]_{22}}{[c]_{22}^2}\bigg)(x_1,x_2,\theta_0)\bigg)_{1\leq i,j\leq d}.}
Then we have the LAMN property if $\Gamma'$ in (\ref{Gamma'-def}) is positive definite almost surely.
\end{enumerate}
}
\end{example}
\begin{remark}
{\color{black} 
If $c^1(x_1,x_2,\theta)=\sqrt{x_2}v$ for a unit vector $v$ in Example~\ref{int-vol-ex}, {\color{black} then} we have $X_t^3=\langle X^1\rangle_t$, and therefore
$X^1$ represents a stock process for a stochastic volatility model and $X^3$ is the integrated volatility process.
If we observe daily stock {\color{black} prices} and realized volatility calculated from high-frequency data, {\color{black} then}
we can regard it as an approximation of the integrated volatility process.
{\color{black} Even though} $c^1=\sqrt{x_2}$ does not {\color{black} satisfy} our assumptions {\color{black} because} $\partial_xc$ is not bounded,
we can approximate this function by setting $c^1$ as a positive-valued smooth function satisfying $c^1(x_1,x_2,\theta)=\sqrt{x_2}$ on $\{|x_2|\geq \epsilon\}$
for small $\epsilon>0$.}
\end{remark}
}



\section{Malliavin calculus and the $L^2$ regularity condition}\label{Mcal-section}

In this section, we {\color{black} show} how to check (A1)--(A5) in Section~\ref{L2regu-subsection} under (B1)--({\color{black} B5}).
The equations for density derivatives in Proposition~\ref{delLogP-eq} are crucial for the proof.
From these equations, we obtain Proposition~\ref{delp-p-est-prop} and Lemma~\ref{root-p-smooth-lemma}, which are necessary for checking (A1).

For a matrix $A$, {\color{black} $|A|$ denotes the Frobenius norm}, $|A|=\sqrt{\sum_{ij}|[A]_{ij}|^2}$.
Let 
\begin{equation*}
L^\theta(V)={\color{black} \sum_{k,k'}[K_j^{-1}(\theta)]_{k,k'} D_j[F_{n,\theta,j}]_{k'}[V]_k}
\end{equation*}
for a vector {\color{black} $V\in \mathbb{R}^{k_j-k_{j-1}}$}. 

The following proposition is essentially from Proposition~4.1 in~\cite{gob01} and Theorem~5 in~\cite{glo-gob08}.
To check (A1), we {\color{black} need} an equation for $\partial_\theta^2p_j$.
\begin{as}
The proof is {\color{black} given in} Section~\ref{Mcal-proof-section} {\color{black} of} the supplementary {\color{black} material}~\cite{fuk-ogi20s}.
\end{as}
\begin{arXiv}
{\color{black} The proof is left to Section~\ref{Mcal-proof-section} in the appendix.}
\end{arXiv}
\begin{proposition}\label{delLogP-eq}
Assume (B1) and (B2). Then $F_{n,\theta,j}$ admits a density denoted by $p_{j,\bar{x}_{j-1}}(x_j,\theta)$.
Moreover, $p_{j,\bar{x}_{j-1}}(x_j,\cdot ) \in C^2(\Theta)$, 
\begin{equation}\label{log-density-p-eq1}
\partial_\theta p_{j,\bar{x}_{j-1}}(x_j,\theta)=p_{j,\bar{x}_{j-1}}(x_j,\theta)E_j\big[\delta_j(L^\theta(\partial_\theta F_{n,\theta,j}))\big|F_{n,\theta,j}=x_j\big],
\end{equation}
and 
  \EQN{\label{log-density-p-eq2}
   \partial_\theta^2 p_{j,\bar{x}_{j-1}}(x_j,\theta)&=p_{j,\bar{x}_{j-1}}(x_j,\theta)E_j\big[\delta_j(L^\theta(\partial_\theta^2 F_{n,\theta,j}))
   +\delta_j(L^\theta(\mathfrak{A}_j))\big|F_{n,\theta,j}=x_j\big]
  
almost everywhere in $x_j\in \mathbb{R}^{k_j-k_{j-1}}$, 
{\color{black} where $\mathfrak{A}_j=(\delta_j(L^\theta(\partial_\theta F_{n,\theta,j}\partial_\theta {\color{black} [F_{n,\theta,j}]_k})))_k$}.
\end{proposition}
\begin{discuss}
{\color{black}}% {{\color[rgb]{0.8,0,0} この二階微分の表現は非同期LAMNのLemma 3.6とも合っている}
\end{discuss}

\begin{as}
The proof of the following proposition is {\color{black} given in} Section~\ref{Mcal-proof-section} {\color{black} of} the supplementary {\color{black} material}~\cite{fuk-ogi20s}.
\end{as}
\begin{arXiv}
{\color{black} The proof of the following proposition is left to Section~\ref{Mcal-proof-section} in the appendix.}
\end{arXiv}

\begin{proposition}\label{delp-p-est-prop}
Assume (B1) and (B2). Then
\begin{eqnarray}
\sup_{i,j,\bar{x}_{j-1},\theta}E_j[|\partial_{\theta_i}
 p_{j,\bar{x}_{j-1}}/p_{j,\bar{x}_{j-1}}|^41_{\{p_{j,\bar{x}_{j-1}}\neq
 0\}}(F_{n,\theta,j})]^{1/4}&\leq &C\alpha_n{\color{black}}% {\color[rgb]{0.8,0.3,0} \bar{k}_n^2}, \label{delp-p-est1} \\ 
\quad
 \sup_{i,l,j,\bar{x}_{j-1},\theta}E_j[|\partial_{\theta_i}\partial_{\theta_l}
 p_{j,\bar{x}_{j-1}}/p_{j,\bar{x}_{j-1}}|^21_{\{p_{j,\bar{x}_{j-1}}\neq
 0\}}(F_{n,\theta,j})]^{1/2}&\leq &C\alpha_n^2{\color{black}}% {\color[rgb]{0.8,0.3,0} \bar{k}_n^4}. \label{delp-p-est2}
\end{eqnarray}
\end{proposition}

Let $\theta_h=\theta_0+\epsilon_nh$ for $h\in \mathbb{R}^d$,
  \EQNN{\mathcal{E}^1_j(x_j,\theta)&=\mathcal{E}_j^1(x_j,\theta,\bar{x}_{j-1})=(E_j[\delta_j(L^\theta(\partial_{\theta_i}F_{n,\theta,j,\bar{x}_{j-1}}))|F_{n,\theta,j,\bar{x}_{j-1}}=x_j])_{i=1}^d, \\
   \mathcal{E}^2_j(x_j,\theta)&=(E_j[\delta_j(L^\theta(\partial_{\theta_i}\partial_{\theta_l}F_{n,\theta,j}))
   +\delta_j(L^\theta({\color{black} \mathfrak{A}_j}))|F_{n,\theta,j}=x_j])_{i,l=1}^d. 
  }
{\color{black}}% {\color[rgb]{0.8,0.3,0} We set the conditional expectations equal to zero when $p_j(x_j,\theta)=0$.
\begin{discuss}
{\color{black}}% {{\color[rgb]{0.8,0,0} そのようにできるということは任意のボレル集合$A$に対して
\begin{equation*}
E[E[\delta|X^\theta=\bar{x}_j]|_{y=X^\theta}1_A(X^\theta)]=E[\delta 1_A(X^\theta)]
\end{equation*}
を満たせばよいが，左辺は$\int E[\delta|X^\theta=\bar{x}_j]1_A(\bar{x}_j)p_{X^\theta}(\bar{x}_j)d\bar{x}_j$
に等しいから$p_{X^\theta}(\bar{x}_j)=0$の時は任意に選べる．
}
\end{discuss}
Then $\mathcal{E}_j^1(x_j,\theta)$ and $\mathcal{E}_j^2(x_j,\theta)$ are measurable with respect to $\theta$ almost everywhere in $x_j$
because 
\EQQ{[\mathcal{E}_j^1(x_j,\theta)]_i=(\partial_{\theta_i}p_j/p_j)1_{\{p_j\neq 0\}}
\quad {\rm and} \quad [\mathcal{E}_j^2(x_j,\theta)]_{il}=(\partial_{\theta_i}\partial_{\theta_l}p_j/p_j)1_{\{p_j\neq 0\}}.}}
\begin{discuss}
{\color{black}}% {{\color[rgb]{0.8,0,0} $i$と$l$を交換しても$[\mathcal{E}^2_j]_{il}$の値が変わらないことはProp 3.1での構成の仕方を見ればわかる．}
\end{discuss}


\noindent
{\bf Proof of Theorem~\ref{Malliavin-LAMN-thm2}.}

We check (A1)--(A3) in Theorem~\ref{L2regu-thm} by setting 
  \EQQ{p_j(\theta)=p_j(x_j,\theta)=p_{j,\bar{x}_{j-1}}(x_j,\theta).}

For sufficiently large $n$, we have $\{\theta_{th}\}_{t\in[0,1]}\subset \Theta$,
  \EQNN{&E_{\theta_0}\bigg[\sum_{j=1}^{m_n}\int_{N_j}[\sqrt{p_j}(x_j,\theta_h)-\sqrt{p_j}(x_j,\theta_0)]^2dx_j\bigg] \\
   &\quad \leq 2E_{\theta_0}\bigg[\sum_{j=1}^{m_n}\int_{N_j}(p_j(x_j,\theta_h)+p_j(x_j,\theta_0))dx_j\bigg] \to 0  
  }
as $n\to\infty$ by (N1), and
\begin{eqnarray}
&&\int_{{\color{black}}% {\color[rgb]{0.8,0.3,0} N_j^c}}\bigg\{\sqrt{p_j}(x_j,\theta_h)-\sqrt{p_j}(x_j,\theta_0)-\frac{\epsilon_n}{2}h\cdot  {\color{black}}% {\color[rgb]{0,0,0.8} \dot{\xi}_j(\theta_0)}\bigg\}^2dx_j \nonumber \\
&&\quad =\int_{{\color{black}}% {\color[rgb]{0.8,0.3,0} M_j}}\bigg\{\int^1_0\epsilon_n{\color{black}}% {\color[rgb]{0.8,0.3,0} \frac{\partial_\theta p_j}{2\sqrt{p_j}}}(x_j,\theta_{th})dt\cdot h-\epsilon_n{\color{black}}% {\color[rgb]{0.8,0.3,0} \frac{\partial_\theta p_j}{2\sqrt{p_j}}}(x_j,\theta_0)\cdot h\bigg\}^2dx_j \nonumber \\
&&\quad
 =\int_{{\color{black}}% {\color[rgb]{0.8,0.3,0} M_j}}\bigg\{\int^1_0\int^t_0\epsilon_n^2h^\top{\color{black}}% {\color[rgb]{0.8,0.3,0} \partial_\theta \bigg(\frac{\partial_\theta p_j}{2\sqrt{p_j}}\bigg)}(x_j,{\color{black}}% {\color[rgb]{0.8,0.3,0} \theta_{sh}})hdsdt\bigg\}^2dx_j \nonumber \\
&&\quad\leq \epsilon_n^4|h|^4\int^1_0E\bigg[{\color{black}}% {\color[rgb]{0.8,0.3,0} \bigg\lVert\frac{\partial_\theta^2p_j}{2p_j}-\frac{\partial_\theta p_j\partial_\theta p_j^\top}{4p_j^2}\bigg\rVert_{{\rm op}}^21_{\{p_j\neq 0\}}}(F_{n,\theta_{sh},j},\theta_{sh})\bigg]ds. \nonumber 
\end{eqnarray}
Together with Proposition~\ref{delp-p-est-prop}, we have
\begin{equation}
\sum_{j=1}^{m_n}E\bigg[\int_{{\color{black}}% {\color[rgb]{0.8,0.3,0} N_j^c}}\bigg\{\sqrt{p_j}(x_j,\theta_h)-\sqrt{p_j}(x_j,\theta_0)-\frac{\epsilon_n}{2}h\cdot  {\color{black}}% {\color[rgb]{0,0,0.8} \dot{\xi}_j(\theta_0)}\bigg\}^2dx_j\bigg] \to 0,
\end{equation}
which implies (A1).

Moreover, we have (A2) {\color{black} because}
\begin{equation*}
E_{\theta_0}\bigg[{\color{black}}% {\color[rgb]{0.8,0.3,0} \frac{\partial_\theta p_j}{p_j}}1_{\{p_j\neq 0\}}(x_j,\theta_0)\bigg|{\color{black}}% {\color[rgb]{0.8,0.3,0} \mathcal{F}_{j-1}}\bigg]=\int \partial_\theta p_j(x_j,\theta_0)dx_j=0,
\end{equation*}
{\color{black}}% {\color[rgb]{0.8,0.3,0} where $\mathcal{F}_{j-1}$ is the one in Section~\ref{L2regu-subsection}.}

Further, Proposition~\ref{delp-p-est-prop} yields (A3).

\qed

In the following, we prove Theorem~\ref{Malliavin-LAMN-thm}.
To show (A5), we replace $\mathcal{E}_j^1(F_{n,\theta,j},\theta)$ by $(\mathcal{L}_{j,i,\bar{x}_{j-1}}(\tilde{F}_{n,\theta,j}))_{i=1}^d$,
and then we apply (B4). For that purpose, we first estimate {\color{black} the} difference between $K_j$ and $\tilde{K}_j$.
\begin{lemma}\label{tildeK-est-lemma}
Assume (B1)--(B3) and that $\alpha_n\rho_n\bar{k}_n^2\to 0$. Then, for any $1\leq j\leq m_n$ and $p>1$, $\tilde{K}_j(\theta)$ is an invertible matrix almost surely and satisfies
\begin{equation}
\sup_{i,l,j,\bar{x}_{j-1},\theta}\lVert [K_j(\theta)-\tilde{K}_j(\theta)]_{il}\rVert_{2,p} \leq C_p\rho_n,
\quad
{\color{black}}% {\color[rgb]{0.8,0.3,0} \sup_{j,\bar{x}_{j-1},\theta}\lVert \tilde{K}_j^{-1}(\theta)\rVert_{{\rm op}}}\leq
C\alpha_n{\color{black}}% {\color[rgb]{0,0,0.8} \bar{k}_n}
\end{equation}
for sufficiently large $n$.
\end{lemma}

\begin{as}
The proof is {\color{black} given in} Section~\ref{Mcal-proof-section} {\color{black} of} the supplementary {\color{black} material}~\cite{fuk-ogi20s}.
\end{as}
\begin{arXiv}
{\color{black} The proof is left to Section~\ref{Mcal-proof-section} in the appendix.}
\end{arXiv}


\begin{proposition}\label{delLogP-approx-prop}
Assume {\color{black} (B1)--(B3)} and that $\alpha_n\rho_n\bar{k}_n^2\to 0$ as $n\to \infty$. Then 
{\color{black} there exists a positive constant $C$} such that
\begin{equation}\label{delLogP-approx-prop-eq2}
\sup_{i,j,\bar{x}_{j-1},\theta}E_j\bigg[\bigg|\frac{\partial_{\theta_i} p_{j,\bar{x}_{j-1}}}{p_{j,\bar{x}_{j-1}}}1_{\{p_j\neq 0\}}(F_{n,\theta,j},\theta)-\mathcal{L}_{j,i,\bar{x}_{j-1}}(\tilde{F}_{n,\theta,j},\theta)\bigg|^2\bigg]^{1/2}
\leq C\alpha_n^2\rho_n{\color{black}}% {\color[rgb]{0.8,0.3,0} \bar{k}_n^4}
\end{equation}
for sufficiently large $n$.
\end{proposition}


\begin{proof}
{\color{black} For $V\in (\mathbb{D}_j^{1,p})^{k_j-k_{j-1}}$, 
we regard $D_jV=(D_j[V]_l)_l$ as a vector of size $k_j-k_{j-1}$.}
Let ${\bf L}_{j,i}^\theta=\partial_{\theta_i} \tilde{F}_{n,\theta,j}^\top \tilde{K}_j^{-1}(\theta)D_j\tilde{F}_{n,\theta,j}$.
{\color{black} First, we} show that
\begin{equation}\label{tildeL-approx}
\sup_{i,j,\bar{x}_{j-1},\theta}\lVert L^\theta(\partial_{\theta_i} F_{n,\theta,j})-{\bf L}_{j,i}^\theta\rVert_{\mathbb{D}^{1,p}(H_j)}\leq C_p\alpha_n^2\rho_n{\color{black}}% {\color[rgb]{0.8,0.3,0} \bar{k}_n^4}.
\end{equation}

Condition~(B3) and Lemma~\ref{tildeK-est-lemma} yield estimates for 
\EQQ{(\partial_{\theta_i} F_{n,\theta,j}-\partial_{\theta_i} \tilde{F}_{n,\theta,j})^\top K_j^{-1}D_jF_{n,\theta,j}
\quad {\rm and}\quad \partial_{\theta_i} \tilde{F}_{n,\theta,j}^\top\tilde{K}_j^{-1}(D_jF_{n,\theta,j}-D_j\tilde{F}_{n,\theta,j}).}
{\color{black} Because} $K_j^{-1}-\tilde{K}_j^{-1}=\tilde{K}_j^{-1}(\tilde{K}_j-K_j)K_j^{-1}$, 
we also obtain an estimate for $\partial_{\theta_i} \tilde{F}_{n,\theta,j}^\top(K_j^{-1}-\tilde{K}_j^{-1})D_jF_{n,\theta,j}$.
Then we have (\ref{tildeL-approx}).
\begin{discuss}
{\color{black}}% {{\color[rgb]{0.8,0,0} $|\partial_\theta F_{n,\theta,j}|\lVert \tilde{K}_j^{-1}\rVert_{{\rm op}}|(\tilde{K}_j-K_j)K_j^{-1}D_jF_{n,\theta,j}|\leq C\bar{k}_n^{1/2}\alpha_n\bar{k}_n\cdot \bar{k}_n^{1/2}\rho_n\bar{k}_n^2\alpha_n$.}
\end{discuss}

Moreover, Proposition~1.3.3 in Nualart~\cite{nua06} and (B3) yield
  \EQN{\label{deltaTildeL-est}
   \delta_j({\bf L}_{j,i}^\theta)&=\partial_{\theta_i} \tilde{F}_{n,\theta,j}^\top\tilde{K}_j^{-1}\delta_j(D_j\tilde{F}_{n,\theta,j})
   -{\rm tr}(\tilde{K}_j^{-1}\langle D_j\partial_{\theta_i} \tilde{F}_{n,\theta,j},D_j\tilde{F}_{n,\theta,j}\rangle_{H_j}) \\
   &=\tilde{F}_{n,\theta,j}^\top B_{j,i,\theta}^\top\tilde{K}_j^{-1}\tilde{F}_{n,\theta,j}
   -{\rm tr}(\tilde{K}_j^{-1}B_{j,i,\theta}\tilde{K}_j)=\mathcal{L}_{j,i,\bar{x}_{j-1}}(\tilde{F}_{n,\theta,j},\theta).
  }
\begin{discuss}
{\color{black}}% {{\color[rgb]{0.8,0,0} \begin{equation*}
\sum_{k,n}\langle D^kB^{m,n}\tilde{F}^n,D^k\tilde{F}^l\rangle=\sum_nB^{mn}K_{n,l}=(BK)_{ml}.
\end{equation*}
}
\end{discuss}

Together with Proposition~\ref{delLogP-eq}, we have
  \EQNN{&\frac{\partial_{\theta_i} p_j}{p_j}1_{\{p_j\neq 0\}}(F_{n,\theta,j})-\mathcal{L}_{j,i,\bar{x}_{j-1}}(\tilde{F}_{n,\theta,j}, \theta) \\
   &\quad =E_j[\delta_j(L^\theta(\partial_{\theta_i} F_{n,\theta,j})-{\bf L}_{j,i}^\theta)|F_{n,\theta,j}] +E_j[r_i||F_{n,\theta,j}]-r_i,
  }
where $r_i=\mathcal{L}_{j,i,\bar{x}_{j-1}}(\tilde{F}_{n,\theta,j},\theta)-\mathcal{L}_{j,i,\bar{x}_{j-1}}(F_{n,\theta,j},\theta)$.
\begin{discuss}
{\color{black}}% {{\color[rgb]{0.8,0,0} Nualart Prop 1.5.4より$\lVert \delta(u)\rVert_{0,p}\leq C_p\lVert u\rVert_{\mathbb{D}^{1,p}(H)}$で証明を見ると$C_p$は$j$によらずにとれる．$E[|r_i|^p]^{1/p}$の評価で$\bar{k}_n^3$がでる．}
\end{discuss}
Then we obtain (\ref{delLogP-approx-prop-eq2}) by (B3), (\ref{tildeL-approx}), {\color{black}}% {\color[rgb]{0.8,0.3,0} and {\color{black} the fact} that
\begin{equation}\label{r-est}
E_j[|r_i|^p]^{1/p}\leq C_p\alpha_n\rho_n\bar{k}_n^3
\end{equation}
for any $p\geq 1$.
}

\end{proof}

{\color{black}}% {\color[rgb]{0.8,0.3,0}
\begin{lemma}\label{root-p-smooth-lemma}
Assume (B1), (B2), and (N2). Then, for any $n\in\mathbb{N}$, $1\leq j\leq m_n$, and $h\in\mathbb{R}^d$ satisfying $\{\theta_{th}\}_{t\in [0,1]}\subset \Theta$, 
the function $\sqrt{p_{j,\bar{x}_{j-1}}} (x_j,\theta_{th})$ is absolutely continuous on $t\in[0,1]$
almost everywhere in $x_j$.
\end{lemma}

\begin{as}
The proof is {\color{black} given in} Section~\ref{Mcal-proof-section} {\color{black} of} the supplementary {\color{black} material}~\cite{fuk-ogi20s}.
\end{as}
\begin{arXiv}
{\color{black} The proof is left to Section~\ref{Mcal-proof-section} in the appendix.}
\end{arXiv}


\begin{lemma}\label{L2-regu-lemma}
Assume (B1)--(B3), (B5), and (\ref{rho-condition}). Then (A1) holds true.
\end{lemma}

\begin{proof}
If (N1) is satisfied, {\color{black} then} the proof of Theorem~\ref{Malliavin-LAMN-thm2} implies (A1). {\color{black} Thus,} we may assume (N2).
We fix $h\in\mathbb{R}^d$ and consider a sufficiently large $n$ so that $\{\theta_{th}\}_{t\in [0,1]}\subset \Theta$.
Thanks to Lemma~\ref{root-p-smooth-lemma}, $\partial_t\sqrt{p_{j,t}}$ exists almost everywhere in $t\in [0,1]$ and $\sqrt{p_{j,1}}-\sqrt{p_{j,0}}=\int^1_0\partial_t\sqrt{p_{j,t}}dt$
almost everywhere in $x_j\in\mathbb{R}^{k_j-k_{j-1}}$.
Moreover, we can see that $\partial_t\sqrt{p_{j,t}}=\partial_tp_{j,t}/(2\sqrt{p_{j,t}})$ when $p_{j,t}\neq 0$ by Proposition~\ref{delLogP-eq}.

For $t\in (0,1)$ such that $\partial_t\sqrt{p_{j,t}}$ exists and $p_{j,t}=0$, we have
\begin{equation*}
\liminf_{s\searrow t}\frac{\sqrt{p_{j,s}}}{s-t}\geq 0 \quad {\rm and} \quad \limsup_{s\nearrow t}\frac{\sqrt{p_{j,s}}}{s-t}\leq 0,
\end{equation*}
which imply $\partial_t\sqrt{p_{j,t}}=0$. Therefore, we obtain	
\begin{equation*}
\sqrt{p_{j,1}}-\sqrt{p_{j,0}}=\int^1_0\frac{\partial_tp_{j,t}}{2\sqrt{p_{j,t}}}1_{\{p_{j,t}\neq 0\}}dt.
\end{equation*}
Then we have
  \EQN{\label{l2-regu-eq1}
   &\sum_{j=1}^{m_n}\int \bigg(\sqrt{p_{j,1}}(x_j)-\sqrt{p_{j,0}}(x_j)-\frac{\partial_\theta p_j\cdot \epsilon_nh}{2\sqrt{p_j}}1_{\{p_j\neq 0\}}(x_j,\theta_0)\bigg)^2dx_j \\
   &\quad =\sum_{j=1}^{m_n}\int \bigg(\int^1_0\frac{\partial_\theta p_j\cdot \epsilon_nh}{2\sqrt{p_j}}1_{\{p_j\neq 0\}}(x_j,\theta_{th})dt-\frac{\partial_\theta p_j\cdot \epsilon_nh}{2\sqrt{p_j}}1_{\{p_j\neq 0\}}(x_j,\theta_0)\bigg)^2dx_j \\
   &\quad \leq \epsilon_n^2|h|^2\sum_{j=1}^{m_n}\int^1_0\int \bigg|\frac{\partial_\theta p_j}{2\sqrt{p_j}}1_{\{p_j\neq 0\}}(x_j,\theta_{th})-\frac{\partial_\theta p_j}{2\sqrt{p_j}}1_{\{p_j\neq 0\}}(x_j,\theta_0)\bigg|^2dx_jdt. 
  }

Let 
\begin{equation*}
\mathfrak{G}_{j,t}=\frac{\partial_\theta p_j}{2\sqrt{p_j}}1_{\{p_j\neq 0\}}(x_j,\theta_{th})-\frac{\sqrt{p_j}}{2}(\mathcal{L}_{j,i,\bar{x}_{j-1}}(x_j,\theta_{th}))_{i=1}^d,
\end{equation*}
then (\ref{r-est}), Proposition~\ref{delLogP-approx-prop}, and (\ref{rho-condition}) yield
  \EQN{\label{l2-regu-eq2}
   &\epsilon_n^2\sum_{j=1}^{m_n}\int^1_0\int |\mathfrak{G}_{j,t}-\mathfrak{G}_{j,0}|^2dx_jdt
   \leq 2\epsilon_n^2\sup_t \bigg(\sum_{j=1}^{m_n}\int (|\mathfrak{G}_{j,t}|^2+|\mathfrak{G}_{j,0}|^2)dx_j\bigg) \\
   &\quad \leq \epsilon_n^2\sup_t \bigg(\sum_{j=1}^{m_n}E_j\bigg[\bigg|\frac{\partial_\theta p_j}{p_j}1_{\{p_j\neq 0\}}
   -(\mathcal{L}_{j,i,\bar{x}_{j-1}})_{i=1}^d\bigg|^2(F_{n,\theta_{th},j},\theta_{th})\bigg]\bigg) \\
   &\quad \leq C\epsilon_n^2m_n(\alpha_n^4\rho_n^2\bar{k}_n^8+\alpha_n^2\rho_n^2\bar{k}_n^6) \to 0
  }
for any $\bar{x}_{j-1}$.

Moreover, {\color{black} because} the function $t\mapsto \sqrt{p_{j,t}}\mathcal{L}_{j,i,\bar{x}_{j-1}}(x_j,\theta_{th})$ is absolutely continuous, we have
  \EQN{\label{l2-regu-eq3}
   &\epsilon_n^2\sum_{j=1}^{m_n}\int^1_0\int \bigg(\frac{\sqrt{p_j}}{2}\mathcal{L}_{j,i}(x_j,\theta_{th})-\frac{\sqrt{p_j}}{2}\mathcal{L}_{j,i}(x_j,\theta_0)\bigg)^2dx_jdt \\
   &\quad =\epsilon_n^2\sum_{j=1}^{m_n}\int^1_0\int \bigg(\int^t_0\bigg(\frac{\sqrt{p_j}}{2}\partial_\theta\mathcal{L}_{j,i}+\frac{\partial_\theta p_j}{4\sqrt{p_j}}\mathcal{L}_{j,i}\bigg)(x_j,\theta_{sh})\cdot \epsilon_nhds\bigg)^2dx_jdt \\
   &\quad \leq C\epsilon_n^4m_n\sup_tE_j\bigg[\bigg|\frac{1}{2}\partial_\theta \mathcal{L}_{j,i}+\frac{\partial_\theta p_j}{4p_j}\mathcal{L}_{j,i}\bigg|^2(F_{n,\theta_{th},j},\theta_{th})\bigg]
  }
for any $1\leq i\leq d$. 

{\color{black} Because} 
 $\partial_\theta \tilde{K}_j^{-1}=-\tilde{K}_j^{-1}\partial_\theta \tilde{K}_j\tilde{K}_j^{-1}$ 
and 
\begin{equation*}
\partial_{\theta_i}\tilde{K}_j
  =\langle \partial_{\theta_i}D_j\tilde{F}_{n,\theta,j},D_j\tilde{F}_{n,\theta,j}\rangle + \langle D_j\tilde{F}_{n,\theta,j},\partial_{\theta_i}D_j\tilde{F}_{n,\theta,j}\rangle
  =B_{j,i,\theta}\tilde{K}_j+\tilde{K}_jB_{j,i,\theta}^\top,
\end{equation*}
\begin{discuss}
{\color{black}}% {{\color[rgb]{0.8,0,0} $F_\theta,G_\theta$がFrechet微分可能なら内積の連続性より
\begin{eqnarray}
h^{-1}(\langle F_{\theta+h},G_{\theta+h}\rangle-\langle F_\theta,G_\theta\rangle)
&=&h^{-1}(\langle F_{\theta+h}-F_\theta,G_{\theta+h}\rangle+\langle F_\theta,G_{\theta+h}-G_\theta\rangle) \nonumber \\
&\to&\langle \partial_\theta F_\theta,G_\theta\rangle+\langle F_\theta,\partial_\theta G_\theta\rangle. \nonumber
\end{eqnarray}
}
\end{discuss}
Lemma~\ref{tildeK-est-lemma} yields
\EQN{\label{del-L-est}
&E_j[|\partial_{\theta_l}\mathcal{L}_{j,i,\bar{x}_{j-1}}(F_{n,\theta,j},\theta)|^2] \\
&\quad \leq CE_j[|F_{n,\theta,j}^\top\partial_{\theta_l}B_{j,i,\theta}^\top \tilde{K}_j^{-1}F_{n,\theta,j}|^2] \\
&\quad \quad +CE_j[|F_{n,\theta,j}^\top B_{j,i,\theta}^\top (B_{j,l,\theta}^\top\tilde{K}_j^{-1}+\tilde{K}_j^{-1}B_{j,l,\theta})F_{n,\theta,j}|^2] \\
&\quad \leq C\alpha_n^2\bar{k}_n^8
}
for any $\theta\in\Theta$.
Moreover, (\ref{r-est}) yields 
  \EQN{\label{L-F-est}
   &E_j[|\mathcal{L}_{j,i,\bar{x}_{j-1}}(F_{n,\theta,j})|^4] \\
   &\quad \leq CE_j[|\mathcal{L}_{j,i,\bar{x}_{j-1}}(F_{n,\theta,j})-\mathcal{L}_{j,i,\bar{x}_{j-1}}(\tilde{F}_{n,\theta,j})|^4]
   +CE_j[|\mathcal{L}_{j,i,\bar{x}_{j-1}}(\tilde{F}_{n,\theta,j})|^4] \\
   &\quad \leq C(\alpha_n\rho_n\bar{k}_n^3)^4+CE_j[|\partial_{\theta_i}\tilde{F}_{n,\theta,j}^\top \tilde{K}_j^{-1}\tilde{F}_{n,\theta,j}|^4]
   \leq C(\alpha_n\rho_n\bar{k}_n^3+\alpha_n\bar{k}_n^2)^4
  }
for any $\theta\in\Theta$.

(\ref{l2-regu-eq3})--(\ref{L-F-est}) and Proposition~\ref{delp-p-est-prop} yield
  \EQNN{&\epsilon_n^2\sum_{j=1}^{m_n}\int^1_0\int \bigg(\frac{\sqrt{p_j}}{2}\mathcal{L}_{j,i,\bar{x}_{j-1}}(x_j,\theta_{th})-\frac{\sqrt{p_j}}{2}\mathcal{L}_{j,i,\bar{x}_{j-1}}(x_j,\theta_0)\bigg)^2dx_jdt  \\
   &\quad \leq C\epsilon_n^4m_n(\alpha_n^2\bar{k}_n^8+\alpha_n^2\bar{k}_n^4(\alpha_n^2\rho_n^2\bar{k}_n^6+\alpha_n^2\bar{k}_n^4)).
  }
The right-hand side converges to zero by (B2) and (\ref{rho-condition}).
Together with (\ref{l2-regu-eq1}) and (\ref{l2-regu-eq2}), we obtain the conclusion.
\end{proof}
}

\noindent
{\bf Proof of Theorem~\ref{Malliavin-LAMN-thm}.}

{\color{black}}% {\color[rgb]{0.8,0.3,0} Thanks to {\color{black} Remark~\ref{Gamma-pd-rem},} Lemma~\ref{L2-regu-lemma}, and the proof of Theorem~\ref{Malliavin-LAMN-thm2},} 
it is sufficient to check (A4) and (A5) {\color{black} under (B1)--(B5)}.
Let $X_j=X_j^{n,\theta_0}$, $\bar{X}_{j-1}=(X_1,\cdots, X_{j-1})$, and
\begin{equation*}
\mathcal{H}_j=E[\mathcal{E}_j^1(\mathcal{E}_j^1)^\top(X_j,\theta_0,\bar{X}_{j-1})|{\color{black}}% {\color[rgb]{0.8,0.3,0} \sigma(\bar{X}_{j-1})}].
\end{equation*}
Then it suffices to show that 
\begin{equation}\label{A4-suff}
\sup_m\bigg(\epsilon_n^2\sum_{j=1}^{m_n}E[|\mathcal{H}_j|]\bigg)<\infty
\end{equation}
and
\begin{equation}\label{A5-suff}
\bigg(\epsilon_n\sum_{j=1}^{m_n}\mathcal{E}_j^1(X_j,\theta_0,\bar{X}_{j-1}),\epsilon_n^2\sum_{j=1}^{m_n}\mathcal{H}_j\bigg)
\overset{d}\to (\Gamma^{1/2}\mathcal{N},\Gamma).
\end{equation}

For sufficiently large $n$, (\ref{Gj-def-eq}) and Proposition~\ref{delLogP-approx-prop} yield
  \EQNN{&E[|\mathcal{E}_j^1(X_j,\theta_0,\bar{X}_{j-1})-G_j^n|^2|\mathcal{G}_{j-1}] \\
   &\quad =E_j[|\mathcal{E}_j^1(F_{n,\theta_0,j},\theta_0,\bar{x}_{j-1})-(\mathcal{L}_{j,i,\bar{x}_{j-1}}(\tilde{F}_{n,\theta_0,j},\theta_0))_{i=1}^d|^2]|_{\bar{x}_{j-1}=\bar{X}_{j-1}} \\
   &\quad \leq C\alpha_n^4\rho_n^2\bar{k}_n^8. 
  }
\begin{discuss}
{\color{black}}% {{\color[rgb]{0.8,0,0} (\ref{Gj-def-eq})より任意のボレル関数$f$に対して
\begin{equation*}
E_j[f(\bar{x}_{j-1},F_{n,\theta,j,\bar{x}_{j-1}},\mathcal{L}_{j,i,\bar{x}_{j-1}}(\tilde{F}_{n,\theta,j,\bar{x}_{j-1}}))]|_{\bar{x}_{j-1}=X_{j-1}^{n,\theta_0}}
=E[f(X_{j-1}^{n,\theta_0},X_j^{n,\theta_0},G_j^n)|\mathcal{G}_{j-1}].
\end{equation*}
(∵$f(x,y,z)=1_A(x)1_B(y,z)$の時はOK. $\pi-\lambda$定理と極限操作より一般の場合もよい)

}
\end{discuss}
Together with (\ref{rho-condition}) {\color{black}}% {\color[rgb]{0.8,0.3,0} and (B4)}, we obtain
  \EQN{\label{del-log-p-Gj-diff-est}
   &E\bigg[\bigg|\epsilon_n\sum_{j=1}^{m_n}\mathcal{E}_j^1(X_j,\theta_0,\bar{X}_{j-1})-\epsilon_n\sum_{j=1}^{m_n}G_j^n\bigg|^2\bigg] \\
   &\quad \leq \epsilon_n^2\sum_{j=1}^{m_n}E\bigg[E\bigg[\bigg|\mathcal{E}_j^1(X_j,\theta_0,\bar{X}_{j-1})-G_j^n\bigg|^2\bigg|\mathcal{G}_{j-1}\bigg]\bigg] \\
   &\quad =O(\epsilon_n^2m_n\alpha_n^4\rho_n^2\bar{k}_n^8) \to 0 
  }
as $n\to \infty$.
\begin{discuss}
{\color{black}}% {{\color[rgb]{0.8,0,0} 
\begin{equation*}
E\bigg[\frac{\partial_\theta p_{j,X_{j-1}^{n,\theta_0}}}{p_{j,X_{j-1}^{n,\theta_0}}}(X_j^{n,\theta_0},\theta_0)\bigg|\mathcal{G}_{j-1}\bigg]
=E_j\bigg[\frac{\partial_\theta p_{j,\bar{x}_{j-1}}}{p_{j,\bar{x}_{j-1}}}(F_{n,\theta_0,j},\theta_0)\bigg]\bigg|_{\bar{x}_{j-1}=X_{j-1}^{n,\theta_0}}=0
\end{equation*}
に注意．
}
\end{discuss}

Let $\mathfrak{F}_j=({\color{black}}% {\color[rgb]{0.8,0.3,0}\mathcal{L}_{j,i,\bar{x}_{j-1}}}(\tilde{F}_{n,\theta_0,j}))_{i=1}^d$, then we have
\begin{equation}\label{gamma-j-eq}
\gamma_j(\bar{x}_{j-1})=E_j[(\mathcal{L}_{j,i,\bar{x}_{j-1}}(\tilde{F}_{n,\theta_0,j})\mathcal{L}_{j,l,\bar{x}_{j-1}}(\tilde{F}_{n,\theta_0,j}))_{il}],
\end{equation}
{\color{black}}% {\color[rgb]{0,0,0.8} and} $\sup_{\bar{x}_{j-1}}E_j[|\mathfrak{F}_j|^2]^{1/2}=O(\alpha_n
\bar{k}_n^2+{\color{black}}% {\color[rgb]{0.8,0.3,0} \alpha_n\rho_n\bar{k}_n^3})$ {\color{black}}% {\color[rgb]{0.8,0.3,0} by (\ref{L-F-est})}. 
Together with (\ref{rho-condition}) {\color{black}}% {\color[rgb]{0.8,0.3,0} and Propositions~\ref{delp-p-est-prop} and~\ref{delLogP-approx-prop}}, we have
  \EQN{\label{gamma-diff}
   &\sup_{\bar{x}_{j-1}}|E_j[\mathcal{E}_j^1(\mathcal{E}_j^1)^\top(F_{n,\theta_0,j},\theta_0,\bar{x}_{j-1})]-\gamma_j(\bar{x}_{j-1})| \\
   &\quad \leq C\sup_{\bar{x}_{j-1}}\bigg(E_j[\mathcal{E}_j^1|^2]^{\frac{1}{2}}E_j[|\mathcal{E}_j^1-\mathfrak{F}_j|^2]^{\frac{1}{2}}
   +E_j[|\mathfrak{F}_j|^2]^{\frac{1}{2}}E_j[|\mathcal{E}_j^1-\mathfrak{F}_j|^2]^{\frac{1}{2}}\bigg) \\
   &\quad =O({\color{black}}% {\color[rgb]{0.8,0.3,0} \alpha_n\bar{k}_n^2\cdot \alpha_n^2\rho_n \bar{k}_n^4+(\alpha_n\bar{k}_n^2+\alpha_n\rho_n\bar{k}_n^3)\alpha_n^2\rho_n\bar{k}_n^4)})
   =o(\epsilon_n^{-2}m_n^{-1}).
  }

Then, (\ref{del-log-p-Gj-diff-est}), (\ref{gamma-diff}), and (B4) yield
\begin{equation*}
\sup_n\bigg(\epsilon_n^2\sum_{j=1}^{m_n}E[|\mathcal{H}_j|]\bigg)
=\sup_n\bigg(\epsilon_n^2\sum_{j=1}^{m_n}E[|\gamma(\bar{X}_{j-1})|]+O(1)\bigg)<\infty
\end{equation*}
and
  \EQNN{
   &\bigg(\epsilon_n\sum_{j=1}^{m_n}\mathcal{E}_j^1(X_j,\theta_0,\bar{X}_{j-1}),\epsilon_n^2\sum_{j=1}^{m_n}\mathcal{H}_j\bigg) \\
   &\quad=\bigg(\epsilon_n\sum_{j=1}^{m_n}G_j^n, \epsilon_n^2\sum_{j=1}^{m_n}\gamma_j({\color{black} \bar{X}_{j-1}})\bigg)+o_p(1)
   \overset{d}\to(\Gamma^{1/2}\mathcal{N},\Gamma). 
  }
\qed


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