\section{Introduction}
  


Motivated by questions about asymptotic behavior in risk analysis, many works have studied large deviation results for sequences of risk measures; see e.g.~\cite{backhoff,eckstein,follmerII,lacker,yan}. 
These new results have inspired the recent work~\cite{kupper}, where it is developed a  structural theory of monetary risk measures with the crucial property of being maxitive. 
Furthermore, as systematically shown in \cite{kupper}, the basic theory of large deviations can be understood within this framework.          
Namely, in \cite{kupper} a \emph{large deviation principle} (LDP) for maxitive monetary risk measures is formulated, and extended versions of Varadhan's lemma and Bryc's theorem are provided for this type of functionals.  
 A 	\emph{monetary risk measure} is a $[-\infty,\infty]$-valued function $\phi$ on the space $B(E)$ of all real-valued Borel measurable functions on a topological space $E$ which satisfies\footnote{
 Here, we adopt the term `monetary risk measure' of F\"{o}llmer and Schied~\cite{follmer} up to a sign change. 
 Often, a monetary risk measure $\phi$ is defined only on the set $B_b(E)$ of all bounded Borel measurable functions. Any function $\phi\colon B_b(E)\to\field{R}$ can be extended to a function $\bar{\phi}\colon B(E)\to [-\infty,\infty]$ preserving the properties (N), (M), (T) by setting $\bar{\phi}(f)=\lim_{m\to\infty}\lim_{n\to\infty}\phi((f \wedge n)\vee m)$. So it is not restrictive to assume that a monetary risk measure is defined on $B(E)$.}
\begin{itemize}
\item[(N)] Normalization: $\phi(0)=0$,
\item[(M)] Monotonicity: $\phi(f)\le\phi(g)$ whenever $f\le g$,
\item[(T)] Translation invariance: $\phi(f+c)=\phi(f)+c$ for all constant $c\in\field{R}$. 
\end{itemize}
Monetary risk measures are  nonlinear expectations in the sense of~\cite{peng}.  
In risk analysis, the space $E$ models all possible states, and $\phi(f)$ quantifies the riskiness of the   loss function $f$.  
A monetary risk measure $\phi$ is said to be \emph{maxitive} if it satisfies 
$$\phi(f\vee g)\le \phi(f)\vee \phi(g)\quad \mbox{ for all }f,g\in B(E).$$  
This property has the interpretation that 
a maxitive monetary risk measure does not compensate losses by large profits in other states (in contrast to expectations which average over states and thus allow for such a
compensation). 
A basic example of a maxitive monetary risk measure is the \emph{asymptotic entropic risk measure}
\[
\phi_{\rm ent}(f):=\underset{n\to\infty}\limsup\frac{1}{n}\log \mathbb{E}[\exp(n f(X_n))],
\]
where $(X_n)_{n\in\mathbb{N}}$ is a sequence of $E$-valued random variables; see~\cite[Examples 2.3(c)]{kupper}.  
A key result in~\cite{kupper} states that, under a tightness assumption\footnote{Also in \cite{kupper} it is  assumed that $E$ is completely regular.}, a maxitive monetary risk measure $\phi$ satisfies the Laplace principle (LP), that is,
\[
\phi(f)=\underset{x\in E}\sup\{f(x)-I(x)\}\quad\mbox{for all }f\in C_b(E),
\]
where $I(\cdot)$ is the rate function given by\footnote{As usual $C_b(E)$ denotes the set of bounded continuous functions $f\colon E\to\field{R}$. 
Recall that $I\colon E\to [0,\infty]$ is a \emph{rate function} if it is lower semicontinouos and not identically $\infty$.} 
\begin{equation}
I(x)=\underset{f\in C_b(E)}\sup\{f(x)-\phi(f)\}.
\end{equation} 
This result amounts to the well-known Bryc's theorem in the case of the asymptotic entropic risk measure $\phi_{\rm ent}$; see e.g.~\cite[Theorem 4.4.2]{dembo}.   
In the present paper we aim to provide sufficient conditions so that the rate function $I(\cdot)$ admits the representation 
\begin{equation}\label{eq:repRate}
I(x)=\underset{f\in\mathcal{H}}\sup\{f(x)-\phi(f)\},
\end{equation} 
for an arbitrarily given set $\mathcal{H}$ of continuous real-valued functionals on $E$.  
The interest of such a representation is that, in applications, often one needs to work on a small class of functionals $\mathcal{H}$ (for instance, all linear continuous functions, all affine continuous functions,...) where the monetary risk measure $\phi$ has a simple representation so that the rate function $I(\cdot)$ can be easily computed. 
For instance, in the case of the asymptotic entropic risk measure $\phi_{\rm ent}$ for the sample means $X_n=\tfrac{1}{n}\sum_{i=1}^n\xi_i$ of a sequence $(\xi_i)_{i\in\mathbb{N}}$ of real-valued i.i.d. random variables, Cram\'{e}r's theorem asserts that the rate function its $I(\cdot)$ attained on the dual space $\field{R}^\ast=\field{R}$, and is given in that case as the convex conjugate of the logarithmic moment generating function $\Lambda(y)=\log\mathbb{E}[e^{y\xi_1}]$. 

The main result of the present paper shows that, under the above mentioned tightness assumption, the rate function $I(\cdot)$ admits the representation \eqref{eq:repRate} whenever $x\in E$ satisfies the separation condition
\begin{equation}\label{eq:exposingConf}
f(y) - \phi^\ast_{\mathcal{H}}(y)  <  f(x) -\phi^\ast_{\mathcal{H}}(x)\quad\mbox{ for all }y\neq x,
\end{equation} 
where $\phi^\ast_{\mathcal{H}}(y):=\sup_{f\in\mathcal{H}}\{f(y)-\phi(f)\}$. 
In turn, we prove that, if the class of points $x\in E$ satisfying \eqref{eq:exposingConf} is rich enough, then $\phi$ has the representation
\begin{equation}\label{eq:repPhi}
\phi(f)=\underset{x\in E}\sup\{f(x)-\phi^\ast_{\mathcal{H}}(x)\}\quad\mbox{for all }f\in C_b(E).
\end{equation}
A further focus shows, in addition, that the representation \eqref{eq:repPhi} extends to certain class of unbounded continuous functionals $f\colon E\to\field{R}$. 
Moreover, the maxitivity property on $\phi$ can be relaxed to a weaker form of maxitivity in line with \cite{kupper2}.  

In large deviations theory,  
 G\"{a}rtner-Ellis theorem provides the LDP for sequences  of random variables with values on a topological vector space $E$ by testing the rate function on the dual space $E^\ast$; see e.g.~\cite{dembo}.  
This well-known result turns out to be a particular instance of the main result of the present paper for the special choice $\mathcal{H}=E^\ast$.     
As pointed out by \cite{dembo}, G\"{a}rtner-Ellis theorem does not cover all cases in which a LDP exists; there are examples for which the LDP holds but it does not follow from this basic result; see~\cite[Remarks(d),~p.~45]{dembo}. 
In contrast, the theorem provided here allows for arbitrary choices of $\mathcal{H}$, covering  situations where G\"{a}rtner-Ellis theorem fails. 
This is illustrated in Example \ref{ex:classical} below. 
Partial results covering this problem can also be found in~\cite{chen0}.   
       
 
 The present approach relies on topological and order properties and, in particular, it is not needed an underlying probability space. 
 We emphasize that the existing proofs of G\"{a}rtner-Ellis theorem rely on probability concepts such as the Radon-Nykodym derivative, see e.g.~\cite[Theorem 4.5.20]{dembo}, which are not needed here.  
 In line with \cite{kupper2,puhalskii}, the machinery is taken from maxitive integration. 
 More specifically, we use the convex integral introduced by Cattaneo~\cite{cattaneo}, which is conceptually related to the idempotent integral in tropical mathematics~\cite{maslov} and can be obtained as a transformation of the Shilkret integral~\cite{shilkret}. 
     
 
 Finally, we remark that, motivated by asymptotic questions in uncertainty modeling, some recent works have studied large deviation results for sequences of sublinear expectations; see \cite{chen,tan} and references therein.   
 In particular, the main result in \cite{tan} is a version of  G\"{a}rtner-Ellis theorem for sequences of sublinear expectations defined on $\field{R}^d$. 
 As an application of the main result of the present paper, we establish a large deviation result for sequences of sublinear expectations on regular Hausdorff topological spaces, which extends to infinite dimensional spaces the main result in~\cite{tan} with the advantage that the rate function is tested on  arbitrary sets of continuous functions.      
 
The paper is organized as follows. 
In Section \ref{sec:preliminaries} we give some preliminaries on maxitive integration. 
In Section \ref{sec:representation} we focus on the integral representation of maxitive monetary risk measures.  
In Section \ref{sec:main} we state and prove the main result of this paper. 
Finally, in Section \ref{sec:LDsublinearExp} we apply the main result to obtain a large deviations result  for sequences of sublinear expectations on regular Hausdorff topological spaces. 
    

\section{Preliminaries on maxitive integration}\label{sec:preliminaries}
Throughout this paper let $E$ be a regular Hausdorff  topological space with Borel $\sigma$-algebra $\mathcal{B}(E)$.   
We always make the convention that $-\infty \cdot 0=0$. 
Then, given a function $f\colon E \to\field{R}\cup \{-\infty\}$, the function $f 1_A - \infty 1_{A^c}$ takes the same values as $f$ on $A\subset E$ and the value $-\infty$ on $A^c$. 


A set function $J\colon \mathcal{B}(E)\to [-\infty,0]$ is said to be a \emph{concentration} if:
\begin{itemize}
\item[(a1)] $J_\emptyset=-\infty$, $J_E=0$,
\item[(a2)] $J_A\le J_B$ whenever $A\subset B$.
\end{itemize}
We say that $J$ is \emph{maxitive} if:
\begin{itemize}
\item[(a3)] $J_{A\cup B}\le J_A\vee J_B$.
\end{itemize}
Denote by $B(E)$ the set of all Borel measurable functions $f\colon E\to \mathbb{R}\cup\{-\infty\}$. 
We define the \emph{convex integral} of $f\in B(E)$ with respect to the concentration $J$ as
\footnote{It is not difficult to show that $\phi_J(f)=\sup_{c\in\mathbb{R}}\left\{c + J_{\{f> c\}}\right\}=\sup_{c\in\mathbb{R}}\left\{c + J_{\{f\ge  c\}}\right\}$, i.e.~we obtain an equivalent definition of $\phi_J(f)$ if the strict inequality in \eqref{eq:maxitiveInt} is replaced by a non-strict inequality.}
\begin{equation}\label{eq:maxitiveInt}
\phi_J(f)=\underset{c\in\mathbb{R}}\sup\left\{c + J_{\{f> c\}}\right\}.
\end{equation}  
The following properties (b1)--(b4) below are provided in \cite{cattaneo} and their proofs easily adapt to the present setting.  
 The proof of (b5) can be found in~\cite[Lemma 4.1]{kupper2}.    
\begin{itemize}
\item[(b1)] $\phi_J(-\infty 1_{A^c})=J_A$ for all $A\in\mathcal{B}(E)$, 
\item[(b2)] $\phi_J(0)=0$,
\item[(b3)] $\phi_J(f+c)=\phi_J(f)+c$ for all constant $c\in \mathbb{R}$,
\item[(b4)] $\phi_J(f)\le \phi_J(g)$ whenever $f\le g$,
\item[(b5)] $\underset{n\to\infty}\lim \phi_J(f \wedge n)=\phi_J(f)$ and  $\underset{n\to\infty}\lim \phi_J(f \vee -n)=\phi_J(f)$.
\end{itemize}
If $J$ is maxitive, then the following hold:
\begin{itemize}
\item[(b6)] $\phi_J$ is maxitive, i.e. $\phi_J(f\vee g)\le \phi_J(f)\vee \phi_J(g)$, 
\item[(b7)] $\phi_J$ is convex, i.e. $\phi_J(\lambda f + (1-\lambda) g)\le \lambda \phi_J(f) + (1-\lambda) \phi_J(g)$ for all $0\le \lambda \le 1$.
\end{itemize}
The proofs of (b6) and (b7) can be found in \cite[Corollary 5]{cattaneo} and \cite[Theorem 7]{cattaneo}, respectively. 
Denote by ${L}(E)$ the set of all lower semicontinuous functions $f\colon E\to \mathbb{R}\cup\{-\infty\}$, and by ${U}(E)$ the set of all upper semicontinuous functions $f\colon E\to \mathbb{R}\cup\{-\infty\}$. 
The following duality bounds were proved in \cite[Theorem 3.4]{kupper2}.\footnote{\cite[Theorem 3.4]{kupper2} deals with functions which are increasing with respect to a given preorder. 
To apply \cite[Theorem 3.4]{kupper2} here, we consider the trivial preoreder, i.e. $x\le y$ whenever $x=y$.} 
\begin{theorem}\label{thm:dualityBounds}
Let $J$ be a concentration, and $I\colon E\to [0,\infty]$ a function. 
Then, the following equivalences hold. 
First, 
\begin{equation}\label{eq:LowerLDP}
-\underset{x\in O}\inf I(x)\le J_O\quad\mbox{ for all open set }O\subset E
\end{equation}
if and only if
\begin{equation}\label{eq:LowerLP}
\phi_J(f)\ge \underset{x\in E}\sup\{f(x)-I(x)\}\quad\mbox{ for all }f\in  {L}(E). 
\end{equation}
Second, 
\begin{equation}\label{eq:UpperLDP}
J_C\le -\underset{x\in C}\inf I(x)\quad\mbox{ for all closed set }C\subset E
\end{equation}
if and only if
\begin{equation}\label{eq:UpperLP}
\phi_J(f)\le \underset{x\in E}\sup\{f(x)-I(x)\}\quad\mbox{ for all }f\in  {U}(E). 
\end{equation}
\end{theorem}


The minimal rate function $I_{\min}\colon E\to [0,\infty]$ associated with a concentration $J$ is defined as
\begin{equation}\label{eq:minRate}
I_{\min}(x):=\underset{f\in {L}(E)}\sup\{f(x)-\phi_J(f)\}.
\end{equation}

\begin{proposition}\label{prop:lowerBounds}
Let $J$ be a concentration and suppose that $I_{\min}$ is defined as in \eqref{eq:minRate}. 
Then, $I_{\min}$ is the smallest mapping $I\colon E\to[0,\infty]$ which satisfies the equivalent inequalities \eqref{eq:LowerLDP} and \eqref{eq:LowerLP}. 
\end{proposition}


The minimal rate function has the following representation; see~\cite[Lemma 3.5]{kupper2}.  
\begin{proposition}\label{prop:representationRate}
Let $J$ be a concentration and suppose that $I_{\min}$ is defined as in~\eqref{eq:minRate}. 
For all $x\in E$ it holds
\[
-I_{\min}(x)=\underset{U\in\mathcal{U}_x}\inf J_U,
\]
where $\mathcal{U}_x$ is a base of open neighborhoods of $x\in E$. 
\end{proposition}
The following notion was introduced in \cite{kupper2}. 
\begin{definition}
A concentration $J$ is said to be \emph{weakly maxitive} if 
\[
J_C\le \vee_{i=1}^N J_{O_i} \mbox{ 
for all $C\subset E$ closed, $O_1,O_2,\ldots,O_N\subset E$ open, $N\in\mathbb{N}$, such that $C\subset\cup_{i=1}^N O_i$.} 
\]
\end{definition}
Notice that every maxitive concentration $J$ is also weakly maxitive. 





\begin{definition}\label{def:tightness}
We say that a concentration $J$ is \emph{tight} is for every $n\in\mathbb{N}$ there exists a compact set $K\subset E$ such that $J_{K^c}<-n$.
\end{definition}

The following result was provided in \cite[Theorem 4.1]{kupper} (see~\cite[Remark 4.2]{kupper}) and \cite[Corollary 3.10]{kupper2} under slightly different assumptions. 
For the sake of completeness, we provide a short proof.  
\begin{proposition}\label{prop:Bryc}
Let $J$ be a concentration and suppose that $I_{\min}$ is defined as in \eqref{eq:minRate}. 
If $J$ is tight and weakly maxitive, then $I_{\min}$ satisfies \eqref{eq:LowerLDP}, \eqref{eq:LowerLP}, \eqref{eq:UpperLDP}, and \eqref{eq:UpperLP}. 
\end{proposition}
\begin{proof}
In view of Theorem \ref{thm:dualityBounds} and Proposition \ref{prop:lowerBounds}, it is enough to show \eqref{eq:UpperLDP}.  
Suppose that $C\subset E$ is closed. 
Fix $\varepsilon>0$. 
Since $J$ is tight, there exists a compact set such that $-J_{K^c}\ge \varepsilon^{-1}$.   
Due to Proposition~\ref{prop:representationRate} and by compactness, there exists $x_1,\ldots,x_N\in K\cap C$ and open sets $U_1,\ldots,U_N$ such that $x_i\in U_i\subset E$ for all $i\in\{1,2,\ldots,N\}$ and
\[
-J_{U_i}\ge (I(x_i)-\varepsilon)\wedge \varepsilon^{-1}\quad\mbox{ for all }i=1,2,\ldots,N.
\]
Set in addition $U_0:=K^c$. 
We have that $C\subset \cup_{i=0}^N U_i$.  
Since $J$ is weakly maxitive, we have
\[
-J_C \ge \wedge_{i=0}^N (-J_{U_i})\ge \wedge_{i=0}^N (I(x_i)-\varepsilon)\wedge \varepsilon^{-1} 
\ge \left(\underset{x\in C}\inf I(x) -\varepsilon\right)\wedge \varepsilon^{-1}.
\]
Letting $\varepsilon \downarrow 0$, we obtain the result. 
\end{proof}





\section{Integral representation of maxitive monetary risk measures}\label{sec:representation} 
A \emph{monetary risk measure} is a function
 $\phi\colon B(E)\to[-\infty,\infty]$ 
satisfying:
\begin{itemize}
\item[(N)] Normalization:  $\phi(0)=0$,
\item[(M)] Monotonicity: $\phi(f)\le\phi(g)$ whenever $f\le g$,
\item[(T)] Translation invariance: $\phi(f+c)=\phi(f)+c$ for all~$c\in\field{R}$. 
\end{itemize} 
We say that a monetary risk measure $\phi$ is \emph{maxitive} if it satifies 
$$\phi(f\vee g)\le \phi(f)\vee \phi(g)\quad \mbox{ for all }f,g\in B(E).$$ 
 Due to (b2)--(b4) in Section \ref{sec:preliminaries} the convex integral \eqref{eq:maxitiveInt} is a monetary risk measure which is maxitive if the corresponding concentration is maxitive.  
In the following, we focus on the converse direction and analyze when a monetary risk measure can be represented as a convex integral.     
Denote by $\bar{B}(E)$ the set of all $f\in B(E)$ such that $f$ is bounded from above. 
The following result was obtained in a slightly different setting in \cite[Corollary 6]{cattaneo}. 
We provide a short proof in the present setting.    
\begin{theorem}\label{thm:repI}
Let $\phi\colon B(E)\to[-\infty,\infty]$ be a maxitive monetary risk measure, and $J_A=\phi(-\infty 1_{A^c})$ for all $A\in\mathcal{B}(A)$. 
Then,  $J$ is a maxitive concentration, and 
\[
\phi(f)=\phi_J(f)
\]
for all $f\in \bar{B}(E)$.
\end{theorem}
\begin{proof}
For every function $f\colon E\to\field{R}\cup\{-\infty\}$ that is bounded from above and every set $A\subset E$, we define
\[
\bar{\phi}(f):=\underset{g\in B(E)\colon f\le g}\inf\phi(g),\quad
\mbox{ and }\quad 
\bar{J}_A:=\bar{\phi}(-\infty 1_{A^c}).
\]
Inpection shows that $\bar{\phi}$ is finitely maxitive and additively homogeneous in the sense of \cite{cattaneo}. 
Then, by \cite[Corollary 6]{cattaneo}, we have that
\[
\bar{\phi}(f)=\underset{c\in\field{R}}\sup\{c + \bar{J}_{\{f\ge c\}}\}
\]
for all $f$ that are bounded from above. 
In particular, for $f\in B(E)$, we have 
\[
{\phi}(f)=\bar{\phi}(f)= \underset{c\in\field{R}}\sup\{c + \bar{J}_{\{f\ge c\}}\}\\
=\underset{c\in\field{R}}\sup\{c + {J}_{\{f\ge c\}}\}=\phi_J(f).
\]
This completes the proof. 
\end{proof}

By relaxing the maxitivity condition it is still possible to represent a monetary risk measure on continuous functions. 
The following notion was introduced in \cite{kupper2}. 
\begin{definition}
A monetary risk measure $\phi\colon B(E)\to [-\infty,\infty]$ is said to be weakly maxitive if 
\[
\phi(f)\le \vee_{i=1}^N \phi(g_i)\mbox{ for all $f\in {U}(E)$, $g_1,g_2,\ldots,g_N\in {L}(E)$, $N\in\mathbb{N}$, such that $f\le \vee_{i=1}^N g_i$.}
\]
\end{definition}

Define ${C}(E)={U}(E)\cap {L}(E)$ and $\bar{C}(E)={C}(E)\cap \bar{B}(E)$.   
The following result was shown in  \cite[Theorem 4.2]{kupper2}.\footnote{To apply \cite[Theorem 4.2]{kupper2} here we consider the trivial preorder, i.e. $x\le y$ whenever $x=y$.} 
\begin{theorem}\label{thm:repIII}
Let $\phi\colon B(E)\to[-\infty,\infty]$ be a weakly maxitive monetary risk measure, and $J_A=\phi(-\infty 1_{A^c})$ for all $A\in\mathcal{B}(A)$. 
Then, $J$ is a weakly maxitive concentration and
\[
\phi(f)=\phi_J(f)
\]
for all function $f\in \bar{C}(E)$.  
\end{theorem}

Given a function $\phi\colon B(E)\to[-\infty,\infty]$, we define the sets 
\[
B_{\phi}(E):=\left\{f\in B(E)\colon \mbox{there exists }t>1\mbox{ such that }\phi(t f)<\infty\right\},
\]
\[
C_{\phi}(E):=\left\{f\in C(E)\colon \mbox{there exists }t>1\mbox{ such that }\phi(t f)<\infty\right\}.
\] 
\begin{lemma}\label{lem:tail}
Let $\phi\colon B(E)\to[-\infty,\infty]$ be a monetary risk measure. 
If $f\in B_{\phi}(E)$, then
\[
\underset{m\to\infty}\lim{\phi}\left(f 1_{\{f\ge m\}}-\infty 1_{\{f<m\}}\right)=\underset{m\to\infty}\lim{\phi}\left(f 1_{\{f> m\}}-\infty 1_{\{f\le m\}}\right)=-\infty.
\]
\end{lemma}
\begin{proof}
Suppose that $f\in B_{\phi}(E)$, and take $t>1$ such that $\phi(t f)<\infty$.  
Fix $m\in\mathbb{N}$ and define $g=\exp(f-m)$. 
Then, by translation invariance and monotonicity, we have
\begin{align*}
-m + \phi\left(f 1_{\{f\ge m\}}-\infty 1_{\{f<m\}}\right)&=\phi\left(-m  + f 1_{\{f\ge m\}}-\infty 1_{\{f<m\}}\right)\\
&=\phi\left(\log(g)1_{\{g\ge 1\}} - \infty 1_{\{g< 1\}}\right)\\
&\le\phi\left(\log(g^t)\right)\\
&=\phi\left(t(f-m)\right)\\
&=-mt + \phi\left(t f\right).
\end{align*}
Therefore, it follows that
\begin{align*}
\underset{m\to\infty}\lim{\phi}\left(f 1_{\{f> m\}}-\infty 1_{\{f\le m\}}\right)&\le 
\underset{m\to\infty}\lim{\phi}\left(f 1_{\{f\ge m\}}-\infty 1_{\{f<m\}}\right)\\
&\le \underset{m\to\infty}\lim\Big(m(1-t) + \phi\left(t f\right)\Big)
=-\infty, 
\end{align*}
where the latter limit is $-\infty$ since $t>1$ and $\phi\left(t f\right)<-\infty$. This completes the proof. 
\end{proof}

We next extend Theorem~\ref{thm:repI} and Theorem~\ref{thm:repIII} to unbounded functions as follows.
\begin{theorem}\label{thm:repII}
Let $\phi\colon B(E)\to[-\infty,\infty]$ be a monetary risk measure, and the concentration $J_A=\phi(-\infty 1_{A^c})$ for all $A\in\mathcal{B}(E)$. Then,
\begin{enumerate}
\item if $\phi$ is maxitive, then $\phi(f)=\phi_J(f)$ for all $f\in B_{\phi}(E)$,
\item if $\phi$ is weakly maxitive, then $\phi(f)=\phi_J(f)$ for all $f\in C_{\phi}(E)$.
\end{enumerate}
\end{theorem}
\begin{proof}
We prove (2). 
Suppose that $\phi$ is weakly maxitive. 
Fix $f\in C_{\phi}(E)$, and $n\in\mathbb{N}$.  
Since
$$f\le (f \wedge n)\vee \left(f1_{\{f> n\}}-\infty 1_{\{f\le n\}}\right).$$
Since $f$ is upper semicontinuous, and the functions in the maximum on the right hand side are lower semicontinuous,  
 it follows from the weak maxitivity of $\phi$ that
\begin{align*}
{\phi}(f)&\le {\phi}\left(f \wedge n\right)\vee {\phi}\left(f 1_{\{f> n\}}-\infty 1_{\{f\le n\}}\right)\\
&={\phi}_J\left(f \wedge n\right)\vee {\phi}\left(f 1_{\{f> n\}}-\infty 1_{\{f\le n\}}\right).
\end{align*}
where we have applied that ${\phi}\left(f \wedge n\right)={\phi}_J\left(f \wedge n\right)$ by Theorem~\ref{thm:repI}. 
On the other hand, 
\[
\phi(f)\ge \phi(f \wedge n)=\phi_J(f \wedge n).
\]
Both things together yield 
\[
\phi_J(f \wedge n)\le \phi(f)\le {\phi}_J\left(f \wedge n\right)\vee {\phi}\left(f1_{\{f> n\}}-\infty 1_{\{f\le n\}}\right).
\]
We have $\lim_{n\to\infty}{\phi}\left(f1_{\{f> n\}}-\infty 1_{\{f\le n\}}\right)=-\infty$ by Lemma \ref{lem:tail}. 
Then, by letting $n\to \infty$, we get that $\phi(f)=\phi_J(f)$.  
\end{proof}




\section{A representation theorem for weakly maxitive monetary risk measures}\label{sec:main}
Throughout this section we consider two monetary risk measures $\underline{\phi},\overline{\phi}\colon B(E)\to [-\infty,\infty]$ which satisfy the following.
\begin{assumption}\label{ass:jointMax}\mbox{}\newline\vspace{-0.5cm}
\begin{enumerate}
\item For every $f\in B(S)$, $\underline{\phi}(f)\le\overline{\phi}(f)$, 
\item $\overline{\phi}$ is weakly maxitive,
\item
$
\underline{\phi}(f)\le \underline{\phi}(g_1)\vee\Big(\vee_{i=2}^N\overline{\phi}(g_i)\Big)
$ 
for all $f\in {U}(E)$, and $g_1,g_2,\ldots,g_N\in {L}(E)$, $N\in\mathbb{N}$, such that $f\le \vee_{i=1}^N g_i$.\end{enumerate}
\end{assumption} 
\begin{remark}
 Assumption \ref{ass:jointMax} covers the case of a single (weakly) maxitive monetary risk measure $\phi$ by taking $\underline{\phi}:=\overline{\phi}:=\phi$. 
 In that case, (1)--(3) are automatically satisfied. 
In the application in next section, we will deal with lower/upper large deviations bounds, this is the reason why we consider a pair of monetary risk measures rather than a single one.  
\end{remark}
We consider the concentrations $\underline{J},\overline{J}\colon \mathcal{B}(E)\to [-\infty,0]$ given by $\underline{J}_A=\underline{\phi}(-\infty 1_{A^c})$ and $\overline{J}_A=\overline{\phi}(-\infty 1_{A^c})$. 
In addition, we denote by $\underline{I},\overline{I}$ the respective minimal rate functions defined as in \eqref{eq:minRate}.  
In the following, let $\mathcal{H}$ be a distinguished nonempty set of continuous real-valued functions on $E$.      
We define the \emph{conjugate} $\overline{\phi}_{\mathcal{H}}^\ast\colon E\to[-\infty,\infty]$ of $\overline{\phi}$ with respect to $\mathcal{H}$ as
$$
\overline{\phi}_{\mathcal{H}}^\ast(x):=\underset{f\in \mathcal{H}}\sup\{f(x)-\overline{\phi}(f)\}. 
$$
\begin{definition} 
We say that $x\in E$ is an $\mathcal{H}$-\emph{exposed point} of $\overline{\phi}_{\mathcal{H}}^\ast$ if there exists $f\in \mathcal{H}$ such that
\[
f(y)-\overline{\phi}_{\mathcal{H}}^\ast(y)<f(x)-\overline{\phi}_{\mathcal{H}}^\ast(x)\quad\mbox{ for all }y\neq x. 
\]
In that case, we say that $f$ is an \emph{exposing function} for $x$. 
We denote by $\mathscr{E}$ the set of all $\mathcal{H}$-exposed  points $x\in E$ of $\overline{\phi}_{\mathcal{H}}^\ast$ which admit an exposing function $f\in \mathcal{H}$ such that 
\begin{equation}\label{eq:niceExp}
\overline{\phi}(f)=\underline{\phi}(f),\quad\mbox{ and }\quad f\in B_{\overline{\phi}}(E).
\end{equation}
\end{definition} 

\begin{remark}
If $x$ is an $\mathcal{H}$-{exposed point}, then $\overline{\phi}_{\mathcal{H}}^\ast(x)<\infty$ and $f(x)>-\infty$. 
\end{remark}

We present the main result of this section.\footnote{  
As usual,  we denote below by ${\rm int}(A)$ and ${\rm cl}(A)$ the topological interior and closure of $A\subset E$, respectively.   }
\begin{theorem}\label{thm:main}
Suppose that $\overline{J}$ is tight.   
Then:
\begin{enumerate}[(i)]
\item For all $x\in\mathscr{E}$, $\underline{I}(x)=\overline{I}(x)=\phi^\ast_{\mathcal{H}}(x)$.  
\item For every closed set $C\subset E$,
\[
\overline{J}_C\le -\underset{y\in C}\inf \overline{\phi}_{\mathcal{H}}^\ast(x).
\]
\item For every open set $O\subset E$,
\[
 -\underset{y\in O\cap\mathscr{E}}\inf \overline{\phi}_{\mathcal{H}}^\ast(x)\le \underline{J}_O.
\]
\item Suppose in addition that 
\begin{equation}\label{eq:condition}
\underset{x\in O}\inf \overline{\phi}_{\mathcal{H}}^\ast(x)=\underset{x\in O\cap\mathscr{E}}\inf \overline{\phi}_{\mathcal{H}}^\ast(x)
\quad\mbox{ for every open set } O\subset E.
\end{equation} 
Then for all $A\in \mathcal{B}(E)$,
\begin{equation}\label{eq:LDP0}
-\underset{x\in {\rm int}(A)}\inf \overline{\phi}_{\mathcal{H}}^\ast(x)\le \underline{J}_A \le \overline{J}_A \le -\underset{x\in {\rm cl}(A)}\inf \overline{\phi}_{\mathcal{H}}^\ast(x). 
\end{equation}
Moreover, 
$\overline{\phi}$ has the representation
\begin{equation}\label{eq:LP0}
\overline{\phi}(f)=\underset{x\in E}\sup\{f(x)-\overline{\phi}_{\mathcal{H}}^\ast(x)\}
\end{equation}
for all $f\in C_{\overline{\phi}}(E)$.   
\end{enumerate}
\end{theorem}

\begin{remark}
The sequence of inequalities \eqref{eq:LDP0} is an analogue for concentrations of the \emph{large deviation principle} (LDP), and the representation \eqref{eq:LP0} is an analogue for monetary risk measures of the \emph{Laplace principle} (LP).  
The equivalence between LDP and LP for monetary risk measures was provided in \cite[Proposition 5.2]{kupper} under the assumption that $\overline{\phi}$ is maxitive. 
In that case, it is shown that $\overline{\phi}$ and $\underline{\phi}$ agree on bounded continuous functions.     
\end{remark}


In order to prove Theorem \ref{thm:main}, we need some preliminary results. 

\begin{lemma}\label{lem:rateMinKappa}
$\overline{\phi}_{\mathcal{H}}^\ast(x)\le \overline{I}(x)\le \underline{I}(x)$ for all~$x\in E$.
\end{lemma}
\begin{proof}
Given $x\in E$, we know from their respective definitions that $\overline{I}(x)\le \underline{I}(x)$. 
Fix now $f\in\mathcal{H}$ and $n\in\mathbb{N}$. 
Since $\overline{\phi}$ is weakly maxitive and $f\wedge n \in \bar{C}(E)$, by Theorem \ref{thm:repIII} we have
\[
{\phi}_{\overline{J}}(f\wedge n)=\overline{\phi}(f\wedge n).
\]  
Then, by monotonicity, we have
\[
{\phi}_{\overline{J}}(f\wedge n)=\overline{\phi}(f\wedge n)\le \overline{\phi}(f). 
\]
Due to (b5), letting $n\to \infty$ results in    
\[
{\phi}_{\overline{J}}(f)\le \overline{\phi}(f). 
\]
Thus, we have
\[
f(x) - \overline{\phi}(f) \le f(x) - {\phi}_{\overline{J}}(f) \le \overline{I}(x).
\]
Since $f\in\mathcal{H}$ was arbitrary, it follows that
\[
\overline{\phi}_{\mathcal{H}}^\ast(x)=\underset{f\in\mathcal{H}}\sup\{f(x)-\overline{\phi}(f)\}\le \overline{I}(x).
\]
The proof is complete.
\end{proof}

\begin{lemma}\label{lem:unifExposed}
Let $K\subset E$ be compact and $x\in E$ an $\mathcal{H}$-{exposed point} of $\overline{\phi}_{\mathcal{H}}^\ast$ with exposing function $f\in \mathcal{H}$. 
Then, for every open set $U\subset E$ such that $x\in U$ there exists an open set $W\subset E$  such that 
\begin{enumerate}
\item $K\cap U^c \subset W$,
\item $\underset{y\in {\rm cl}(W)}\sup\left\{f(y)-\overline{\phi}_{\mathcal{H}}^\ast(y)\right\}<f(x)-\overline{\phi}_{\mathcal{H}}^\ast(x)$.
\end{enumerate}

\end{lemma}
\begin{proof}
For each $\varepsilon>0$, define
\[
V_\varepsilon:=\big\{y\in E \colon f(y)-\overline{\phi}_{\mathcal{H}}^\ast(y)+\varepsilon < f(x)-\overline{\phi}_{\mathcal{H}}^\ast(x) \big\}.
\]
Since the mapping $y\mapsto f(y)-\overline{\phi}_{\mathcal{H}}^\ast(y)$ is upper semicontinuous, we have that $V_\varepsilon$ is open.   
We claim that there exists $\varepsilon>0$ such that $K\cap U^c\subset V_\varepsilon$.  
Indeed, by contradiction assume that for every $\varepsilon>0$ we can pick up $y_\varepsilon\in K\cap U^c$ such that
\[
 f(y_\varepsilon)-\overline{\phi}_{\mathcal{H}}^\ast(y_\varepsilon)+\varepsilon\ge f(x)-\overline{\phi}_{\mathcal{H}}^\ast(x).
\] 
Then, $(y_\varepsilon)_{\varepsilon>0}$ is a net in the compact set $K$.\footnote{Here, $\{\varepsilon\colon \varepsilon>0\}$ is regarded as downwards directed set.} 
We can take a subnet $(y_{\varepsilon_\alpha})$ such that $y_{\varepsilon_\alpha}\to y\in K\cap U^c $. 
Taking the limsup   on $\alpha$, and using that $y\mapsto f(y)-\overline{\phi}_{\mathcal{H}}^\ast(y)$ is upper semicontinuous, we get
\[
 f(y)-\overline{\phi}_{\mathcal{H}}^\ast(y) \ge f(x)-\overline{\phi}_{\mathcal{H}}^\ast(x).
\]  
Besides, we have that $y\neq x$ as $y\in U^c$ and $x\in U$.  
This contradicts that $x$ is an $\mathcal{H}$-exposed point. 
We have that $K\cap U^c$ is a compact set contained in the open set $V_\varepsilon$. 
 Since $E$ regular, we can find an open set $W\subset E$ such that $K\cap U^c \subset{\rm cl}(W)\subset V_\varepsilon$. 
Finally, 
the set $W$ meets the required conditions.   
\end{proof}

\begin{proposition}\label{eq:concentrationExp}
Suppose that $x\in\mathscr{E}$ and $f\in\mathcal{H}$ is an exposing function for $x$ satisfying~\eqref{eq:niceExp}. 
If $\overline{J}$ is tight, then for every open set $U\subset E$ with $x\in U$ it holds  
\[
\overline{\phi}(f 1_U - \infty 1_{U^c})=\overline{\phi}(f)=\underline{\phi}(f)=\underline{\phi}(f 1_U - \infty 1_{U^c}).
\]
\end{proposition}
\begin{proof}
Let $f\in \mathcal{H}$ be an exposing function for $x$ satisfying~\eqref{eq:niceExp}. 
Fix $n\in\mathbb{N}$.  
Since $\overline{J}$ is tight, there exists a compact set $K\subset E$ such that 
\begin{equation}\label{eq:tightness}
\overline{J}_{K^c}<-2 n. 
\end{equation}
Due to Lemma~\ref{lem:unifExposed}, we can find an open set $W\subset E$ such that  
\begin{equation}
\label{eq:unifExposed}
K\cap U^c\subset W,\quad 
\underset{y\in {\rm cl}(W)}\sup\left\{f(y)-\overline{\phi}_{\mathcal{H}}^\ast(y)\right\}<f(x)-\overline{\phi}_{\mathcal{H}}^\ast(x).
\end{equation}
Since 
\begin{align*}
E&=K\cup K^c\\
&\subset  (K\cap U)\cup (K\cap U^c)\cup \Big(K^c\cap\{f < n+1\}\Big)\cup \Big(K^c\cap\{f > n\}\Big)\\
&\subset   U \cup W \cup \Big(K^c\cap\{f < n+1\}\Big)\cup \{f > n\},
\end{align*}
 due to Assumption \ref{ass:jointMax} we have
\begin{align}
\overline{\phi}(f)&=\underline{\phi}(f)\nonumber\\
&\le  \underline{\phi}\left(f 1_U - \infty 1_{U^c}\right)
\vee \overline{\phi}\left(f 1_W - \infty 1_{W^c}\right)\vee \overline{\phi}\left((n+1) 1_{K^c} - \infty 1_K\right)
\vee \overline{\phi}\left(f 1_{\{f > n\}} - \infty 1_{\{f\le n\}}\right).
 \label{eq:mainIne}
\end{align}
In addition, by the definition of $\overline{\phi}^\ast_{\mathcal{H}}$ we have 
\begin{equation}\label{eq:ineBelow}
\overline{\phi}(f)\ge f(x)-\overline{\phi}^\ast_{\mathcal{H}}(x).
\end{equation}
On the other hand, $\overline{J}$ is weakly maxitive and tight. 
Then, due to Theorem~\ref{thm:repII} and Proposition~\ref{prop:Bryc} we have
\begin{align*}
\overline{\phi}\left(f 1_{W} -\infty 1_{W^c}\right)&\le \overline{\phi}\left(f 1_{{\rm cl}(W)} - \infty 1_{{\rm cl}(W)^c} \right)\\
&=\phi_{\overline{J}}\left(f 1_{{\rm cl}(W)} - \infty 1_{{\rm cl}(W)^c} \right)\\
&\le \underset{y\in {\rm cl}(W)}\sup\{f(y)-\overline{I}(y)\}\\
&\le \underset{y\in {\rm cl}(W)}\sup\{f(y)-\overline{\phi}^\ast_{\mathcal{H}}(y)\}\\
&<f(x)-\overline{\phi}^\ast_{\mathcal{H}}(x),
\end{align*}
where we have used that $\overline{\phi}^\ast_{\mathcal{H}}(y)\le \overline{I}(y)$ due to Lemma \ref{lem:rateMinKappa} in the second inequality and \eqref{eq:unifExposed} in the third inequality. 
The last inequality is strict, then, in view of \eqref{eq:ineBelow}, we can drop the second member of the maximum in \eqref{eq:mainIne}, obtaining
\begin{equation}\label{eq:mainIne2}
\overline{\phi}(f)\le  \underline{\phi}\left(f 1_U - \infty 1_{U^c}\right)
\vee  \overline{\phi}\left((n+1) 1_{K^c} - \infty 1_K\right)
\vee \overline{\phi}\left(f 1_{\{f > n\}} - \infty 1_{\{f\le n\}}\right).
\end{equation}
By monotonicity and translation invariance, we get
\begin{align*}
\overline{\phi}\left((n+1) 1_{K^c} - \infty 1_K\right) &=\overline{\phi}\left(- \infty 1_K\right)+n+1\\
&= \overline{J}_{K^c} + n+1\\
& \le -2 n + n + 1=-n + 1,
\end{align*}
where we have used \eqref{eq:tightness} in the last inequality. Therefore,
\begin{equation}
\label{eq:limitI}
\underset{n\to\infty}\lim \overline{\phi}\left((n+1) 1_{K^c} - \infty 1_K\right)=-\infty.
\end{equation}
On the other hand,  $f\in B_{\overline{\phi}}(E)$ by \eqref{eq:niceExp}. Then, applying Lemma~\ref{lem:tail} we have 
\begin{equation}
\label{eq:limitII}
\lim_{n\to\infty}\overline{\phi}\left(f 1_{\{f > n\}} - \infty 1_{\{f\le n\}}\right)=-\infty.
\end{equation}
Consequently, letting $n\to\infty$ in \eqref{eq:mainIne2} results in
\[
\overline{\phi}(f)\le \underline{\phi}\left(f 1_U - \infty 1_{U^c}\right).
\] 
Finally, by monotonicity we have 
$$\underline{\phi}\left(f 1_U - \infty 1_{U^c}\right)\le \overline{\phi}\left(f 1_U - \infty 1_{U^c}\right)\le  \overline{\phi}(f)\le \underline{\phi}\left(f 1_U - \infty 1_{U^c}\right),$$ 
and the desired equalities follow. 
\end{proof}



We now turn to the proof of Theorem \ref{thm:main}.

\begin{proof} 
Let $x\in\mathscr{E}$, and take an exposing function $f\in\mathcal{H}$ for $x$ satisfying  \eqref{eq:niceExp}.    
Fix an open neighborhood $U$ of $x$ and $\varepsilon>0$.   
Since $f$ is upper semicontinuous, we can find an open neighborhood $V\subset  U$ of $x$ such that 
\begin{equation}\label{eq:continuity}
f(y)<f(x)+\varepsilon\quad\mbox{ for all }y\in V.
\end{equation}
Due to Proposition~\ref{eq:concentrationExp},  
\begin{align*}
\overline{\phi}(f)&=\underline{\phi}(f 1_{V} - \infty 1_{V^c})\\
&\le \underline{\phi}\big((f(x)+\varepsilon)1_V - \infty 1_{V^c}\big) 
=\underline{J}_V + f(x)+\varepsilon\le \underline{J}_U + f(x)+\varepsilon.  
\end{align*}
On the other hand, from the definition of $\phi^\ast_{\mathcal{H}}$ we have that 
\[
\overline{\phi}(f)\ge f(x)-\phi^\ast_{\mathcal{H}}(x).
\]
Combining both things, we get
\[
-\phi^\ast_{\mathcal{H}}(x) \le \underline{J}_U + \varepsilon. 
\] 
Letting $\varepsilon \downarrow 0$, we obtain
\[
-\phi^\ast_{\mathcal{H}}(x) \le \underline{J}_U. 
\] 
Since $U$ was arbitrary, it follows from Proposition~\ref{prop:representationRate} that
\[
\phi^\ast_{\mathcal{H}}(x) \ge \underline{I}(x).
\]  
Finally, we have by Lemma~\ref{lem:rateMinKappa} that $\phi^\ast_{\mathcal{H}}(x)\le\overline{I}(x)\le \underline{I}(x)$, obtaining (i).


 Suppose that $C\subset E$ is closed. 
Since $\overline{J}$ is tight and weakly maxitive, by Proposition~\ref{prop:Bryc} we have that
\[
\overline{J}_C\le -\underset{x\in C}\inf \overline{I}(x) \le -\underset{x\in C}\inf \phi^\ast_{\mathcal{H}}(x),
\]
where we have used that $\phi^\ast_{\mathcal{H}}(x)\le \overline{I}(x)$. This proves (ii).  

Suppose now that $O\subset E$ is open. 
It follows from Proposition \ref{prop:lowerBounds} that
\begin{align*}
\underline{J}_O &\ge -\underset{x\in O}\inf \underline{I}(x)\\
&\ge -\underset{x\in O\cap\mathscr{E}}\inf \underline{I}(x)\\ 
&= -\underset{x\in O\cap\mathscr{E}}\inf \phi^\ast_{\mathcal{H}}(x), 
\end{align*} 
where we have used that $\underline{I}(x)=\phi^\ast_{\mathcal{H}}(x)$ for all $x\in\mathscr{E}$ by Proposition (i).~Then, (iii) follows. 

Finally, \eqref{eq:LDP0} is a consequence of (ii) and (iii) taking into account \eqref{eq:condition}. 
Due to Theorem \ref{thm:dualityBounds}, we also have
\[
\phi_{\overline{J}}(f)=\underset{x\in E}\sup\{f(x)-\phi^\ast_{\mathcal{H}}(x)\}
\] 
for all $f\in C(E)$. 
By Theorem \ref{thm:repII} we have that $\phi_{\overline{J}}(f)=\overline{\phi}(f)$ for all $f\in C_{\overline{\phi}}(E)$, and in turn \eqref{eq:LP0} follows. \end{proof}


\begin{examples}\label{examples:linear}
\mbox{}\newline\vspace{-0.5cm}
\begin{enumerate}
 \item \textbf{Topological vector spaces}: Every topological vector space is regular. 
 Then, Theorem \ref{thm:main} applies to (Hausdorff) topological  vector spaces. 
 In that case, we can consider $\mathcal{H}:=E^\ast$, the set of all lineal continuous real-valued functions on $E$. 
 Define $\Lambda:=\overline{\phi}|_{E^\ast}$.  
 Since $\overline{\phi}$ is weakly maxitive due to Assumption \ref{ass:jointMax}, we have that $\overline{\phi}$ is maxitive on the set $C(E)$. 
It follows from \cite[Proposition 2.1]{kupper} that $\overline{\phi}$ is convex on $C(E)$ and, in particular, $\Lambda$ is a convex function.  
 Then, we have that $\Lambda^\ast:=\phi^\ast_{E^\ast}$ is the convex conjugate of $\Lambda$.     
 \item \textbf{Finite dimension}:
Suppose now that $E=\field{R}^d$ and $\mathcal{H}=(\field{R}^d)^\ast=\field{R}^d$.  
Consider the conditions below.  
\begin{enumerate}
\item $\Lambda(y)=\underline{\phi}(y)=\overline{\phi}(y)$ for $y\in \field{R}^d$,
\item $0$ belongs to the topological interior of $\{y\in\field{R}^d\colon \overline{\phi}(y)<\infty\}$,
\item $\Lambda$ is lower semicontinuous,
\item $\Lambda$ is \emph{essentially smooth} in the sense of \cite[Definition 2.3.5]{dembo}.
\end{enumerate}
Then, under the conditions (a)--(c) above, the condition \eqref{eq:condition}  in Theorem \ref{thm:main} holds. 
This is proved by following word by word the argumentation in the proof of (c) in \cite[Theorem 2.3.6]{dembo}. 
We conclude that, in the present situation, we can replace \eqref{eq:condition} in Theorem \ref{thm:main} by the conditions (a)--(c) above.  
\end{enumerate}
\end{examples}


\section{Large deviation principle for sequences of sublinear expectations}\label{sec:LDsublinearExp}
We finally apply Theorem \ref{thm:main} to study large deviations for sequences of sublinear expectations.  
Denote by ${B}_+(E)$ the set of all Borel measurable functions $f\colon E\to [0,\infty)$.   
A function $\mathcal{E}\colon {B}_+(E)\to [0,\infty]$ is called a \emph{sublinear} expectation if:
\begin{enumerate}
\item $\mathcal{E}(c)=c$ for all constant $c\ge 0$,
\item $\mathcal{E}(f)\le\mathcal{E}(g)$ whenever $f\le g$,
\item $\mathcal{E}(f+g)\le \mathcal{E}(f)+\mathcal{E}(g)$,
\item $\mathcal{E}(a f)=a\mathcal{E}(f)$ for all constant $a\ge 0$.
\end{enumerate}
A functional which satisfies the properties (1)-(4) is also called upper expectation in robust statistics~\cite{huber}, upper coherent prevision in the theory of imprecise probabilities \cite{walley}, or (up to a sign change) coherent risk measure in mathematical finance \cite{delbaen}.
\begin{example}
Suppose that $X$ is an $E$-valued random variable defined on a probability space $(\Omega,\mathcal{F},\mathbb{P})$. 
Consider a nonempty set $\mathcal{P}$ of probability measures on $\mathcal{F}$. 
Then, the mapping $\mathcal{E}\colon {B}_+(E)\to [0,\infty]$  given by
\[
\mathcal{E}(f):=\underset{Q\in\mathcal{P}}\sup\mathbb{E}_Q[f(X)]
\] 
is a sublinear expectation.
\end{example}



In the following, we consider a sequence $(\mathcal{E}_n)_{n\in\mathbb{N}}$ of sublinear expectations. 
We define the lower/upper \emph{asymptotic entropic risk measures} $\underline{\phi},\overline{\phi}\colon B(E)\to\overline{\field{R}}$ as
\[
\underline{\phi}(f):=\underset{n\to\infty}\liminf \frac{1}{n}\log\mathcal{E}_n(e^{n f}),
\quad
\overline{\phi}(f):=\underset{n\to\infty}\limsup \frac{1}{n}\log\mathcal{E}_n(e^{n f}).
\]
Straightforward inspection shows that $\underline{\phi}$ and $\overline{\phi}$ are monetary risk measures. 
The following lemma is well-known in large deviations theory and often referred to as the principle of the largest term; see e.g.~\cite[Lemma 1.2.15]{dembo} for \eqref{eq:largestTermUp} and \cite[Exercise
14.8]{rassoul} for \eqref{eq:largestTermLow}. 
\begin{lemma}\label{lem:largestTerm}
Suppose that $(a_n^1)_{n\in\mathbb{N}},(a_n^2)_{n\in\mathbb{N}},\ldots,(a_n^N)_{n\in\mathbb{N}}$ are $[0,\infty]$-valued sequences, then
\begin{equation}\label{eq:largestTermUp}
\underset{n\to\infty}\limsup \frac{1}{n}\log \sum_{i=1}^N a_n^i\le \vee^N_{i=1} \underset{n\to\infty}\limsup \frac{1}{n}\log a_n^i,
\end{equation}
\begin{equation}\label{eq:largestTermLow}
\underset{n\to\infty}\liminf \frac{1}{n}\log \sum_{i=1}^N a_n^i\le  \Big(\underset{n\to\infty}\liminf \frac{1}{n}\log a_n^1\Big) \vee\Big(\vee_{i=2}^N \underset{n\to\infty}\limsup \frac{1}{n}\log a_n^i\Big).   
\end{equation}
\end{lemma}  

As a consequence, we have. 
\begin{lemma}\label{lem:joinMax}
$\underline{\phi},\overline{\phi}$ satisfy Assumption \ref{ass:jointMax}.
\end{lemma}
\begin{proof}
We prove that
\[
\overline{\phi}(f)\le \vee_{i=1}^N\overline{\phi}(g_i)
\]
 for $f\in {U}(E)$, and $g_1,g_2,\ldots,g_n\in {L}(E)$ with $f\le \vee_{i=1}^n g_i$. 
 Indeed, as a consequence of Lemma \ref{lem:largestTerm}, we have 
\begin{align*}
\overline{\phi}(f)&\le \underset{n\to\infty}\limsup \frac{1}{n}\log\mathcal{E}_n\left(e^{n (\vee_{i=1}^N g_i)}\right)\\
&\le \underset{n\to\infty}\limsup\frac{1}{n}\log \mathcal{E}_n\left( \sum_{i=1}^N e^{n g_i}\right) \\
&\le \underset{n\to\infty}\limsup\frac{1}{n}\log \sum_{i=1}^N\mathcal{E}_n\left(  e^{n g_i}\right) \\
&\le \vee_{i=1}^N \underset{n\to\infty}\limsup\frac{1}{n}\log \mathcal{E}_n\left(  e^{n g_i}\right) \\
&=\vee_{i=1}^N\overline{\phi}(g_i).
\end{align*}
\end{proof}
Adopting the usual terminology of standard large deviations theory \cite{dembo}, we introduce the following: 
\begin{itemize}
\item We say that the sequence $(\mathcal{E}_n)_{n\in\mathbb{N}}$  is \emph{exponentially tight} if for every $n\in\mathbb{N}$ there exists a compact set $K\subset E$ such that 
\[
\underset{n\to\infty}\limsup \frac{1}{n}\log\mu_n(K^c)<-n.
\] 
Here, $\mu_n\colon\mathcal{B}(E)\to[0,1]$ is the capacity associated with $\mathcal{E}_n$ which is given by $\mu_n(A)=\mathcal{E}_n(1_A)$.  
\item We say that $(\mathcal{E}_n)_{n\in\mathbb{N}}$ satisfies the \emph{large deviation principle} (LDP) with rate function $I\colon E\to[0,\infty]$ if  
\begin{align*}
-\underset{x\in {\rm int}(A)}\inf I(x)\le &\liminf_{n\to\infty}\frac{1}{n}\log\mu_n(A)\\ 
\le &\limsup_{n\to\infty}\frac{1}{n}\log\mu_n(A) \le -\underset{x\in {\rm cl}(A)}\inf I(x)
\end{align*}
 for all $A\in\mathcal{B}(E)$.
\item  We say that $(\mathcal{E}_n)_{n\in\mathbb{N}}$ satisfies the \emph{Laplace principle} (LP) with rate function $I\colon E\to[0,\infty]$ if 
\begin{equation}\label{eq:LP2}
\underset{n\to\infty}\limsup\frac{1}{n}\log\mathcal{E}_n(e^{n f})
=\underset{x\in E}\sup\{f(x)-I(x)\}
\end{equation}
for all $f\in C_{\overline{\phi}}(E)$.  
\end{itemize}

Notice that the exponential tightness of $(\mathcal{E}_n)_{n\in\mathbb{N}}$ is equivalent to the tightness of $\overline{J}$ in the sense of Definition \ref{def:tightness}. 

We turn next to the main result of this section. 
 Let $\mathcal{H}$ be a distinguished nonempty set of continuous real-valued functions on $E$.    	
 Denote by $\mathscr{E}$ the set of all $\mathcal{H}$-exposed points of $\overline{\phi}_{\mathcal{H}}^\ast$ which admit  an exposing function $f\in \mathcal{H}$ such that
\[
\underline{\phi}(f)=\overline{\phi}(f)=\underset{n\to\infty}\lim\frac{1}{n}\log\mathcal{E}_n(e^{n f})\quad\mbox{and}\quad f\in B_{\overline{\phi}}(E).  
\]
In view of Lemma \ref{lem:joinMax}, we have the following result as a direct consequence of Theorem \ref{thm:main}. 
\begin{theorem}\label{thm:sublinear}
Suppose that the sequece $(\mathcal{E}_n)_{n\in\mathbb{N}}$ of sublinear expectations is exponentially tight.  
Then: 
\begin{itemize}
\item[(i)] For every closed set $C\subset E$, 
\[
\limsup_{n\to\infty}\frac{1}{n}\log\mu_n(C)\le -\underset{y\in C}\inf \overline{\phi}_{\mathcal{H}}^\ast(x).
\]
\item[(ii)] For every open set $O\subset E$,
\[
 -\underset{y\in O\cap\mathscr{E}}\inf \overline{\phi}_{\mathcal{H}}^\ast(x)\le \liminf_{n\to\infty}\frac{1}{n}\log\mu_n(O).
\]
\item[(iii)]
Suppose that 
\begin{equation}\label{eq:richExpPoints}
\underset{x\in O}\inf \overline{\phi}_{\mathcal{H}}^\ast(x)=\underset{x\in O\cap{\mathscr{E}}}\inf \overline{\phi}_{\mathcal{H}}^\ast(x)\quad\mbox{for all }O\subset E\mbox{ open.}
\end{equation}
Then, $(\mathcal{E}_n)_{n\in\mathbb{N}}$ satisfies the LDP and the LP with rate function $\overline{\phi}_{\mathcal{H}}^\ast\colon E\to [0,\infty]$. 
\end{itemize}
\end{theorem}

In the special case when $\mathcal{H}=E^\ast$ is the dual space of a topological vector space $E$ Theorem \ref{thm:sublinear} above amounts to the well-known G\"{a}rtner-Ellis theorem; see \cite[Theorem 4.5.20]{dembo} and \cite[Theorem 3.1]{tan} for a version for sublinear expectations.  
We have that Theorem \ref{thm:sublinear} above is more flexible as it allows for arbitrary choices for $\mathcal{H}$.     
In the following example taken from \cite{dembo} we show a case where the LDP and LP follow from Theorem \ref{thm:sublinear} above, but it is not covered by G\"{a}rtner-Ellis theorem.  
\begin{example}\label{ex:classical}
For each $n\in\mathbb{N}$, let $X_n$ be a real-valued random variable with exponential distribution of parameter $n$, i.e., $\mathbb{P}(X_n\le x)=1-e^{-nx}$ for $x\ge 0$. 
Consider the lower/upper asymptotic entropic risk measures 
\[
\underline{\phi}(f)=\underset{n\to\infty}\liminf \frac{1}{n}\log\mathbb{E}_{\mathbb{P}}(e^{n f(X_n)}),
\quad
\overline{\phi}(f)=\underset{n\to\infty}\limsup \frac{1}{n}\log\mathbb{E}_{\mathbb{P}}(e^{n f(X_n)}).
\]
For every $m\in\mathbb{N}$, define the compact set $K_m:=[0,m]$. 
Then, $\overline{J}_{K_m^c}=-m$. Hence, $(X_n)_{n\in\mathbb{N}}$ is exponentially tight. 
Consider $\mathcal{H}_1=\field{R}^\ast=\field{R}$ as in G\"{a}rtner-Ellis theorem \cite[Theorem 4.5.20]{dembo}.  
In that case, for every $y\in\mathcal{H}_1=\field{R}$,
\[
\underline{\phi}(y)=\overline{\phi}(y)=\underset{n\to\infty}\lim\frac{1}{n}\log \int_0^\infty n e^{n y x - n x} d x = \begin{cases}0, & \mbox{ if }y<1,\\ \infty, & \mbox{ if }y\ge 1.\end{cases} 
\]
Hence, 
\[
\overline{\phi}^\ast_{\mathcal{H}_1}(x)=\underset{y\in \field{R}}\sup\{ y x - \overline{\phi}(y)\}=
\begin{cases}
x, & \mbox{ if }x\ge 0,\\
\infty,& \mbox{ if }x< 0.
\end{cases}
\]
The only exposed point of $\overline{\phi}^\ast_{\mathcal{H}_1}$ is $0$. Then, for every open set $O\subset\field{R}$ that does not contain the origin, the classical G\"{a}rtner-Ellis theorem only gives a trivial lower bound $-\inf_{y\in O\cap \{0\}}\overline{\phi}^\ast_{\mathcal{H}_1}(y)=-\infty$, and the condition the condition~\eqref{eq:richExpPoints} is not satisfied. 

Now, for every $a\ge 0$, consider the set of continuous functions 
\[
\mathcal{H}_2:=\left\{ f_a  \colon a\ge 0\right\},
\] 
where
$$f_a(x):=
\begin{cases}
-2(x-a), & \mbox{ if }a\le x,\\
2(x-a), & \mbox{ if }0\le x<a,\\
-2a^2(x+1/a), & \mbox{ if }-1/a\le x<0,\\
0, & \mbox{ if }x< -1/a.
\end{cases}
$$
For every $a\ge 0$, we have
\begin{align*}
\underline{\phi}(f_a)&=\overline{\phi}(f_a)=\underset{n\to\infty}\lim \frac{1}{n}\log \int_0^\infty n e^{n f_a(y)-n y} d y\\
&=\underset{n\to\infty}\lim \frac{1}{n}\log \left(\int_0^a n e^{2 n (y-a)-n y} d y + \int_a^\infty n e^{-2 n (y-a)-n y} d y\right)\\
&=\underset{n\to\infty}\lim \frac{1}{n}\log \left( e^{- n a}  \left(4/3 - e^{-na}\right)\right)\\
&=-a + \underset{n\to\infty}\lim \frac{1}{n}\log \left(4/3 - e^{-na}\right)\\
&=-a.
\end{align*}
Then, inspection shows that 
\[
\overline{\phi}^\ast_{\mathcal{H}_2}(x)=\underset{a\ge 0}\sup\{f_a(x) - \overline{\phi}(f_a)\}=
\begin{cases}
x, & \mbox{ if }x\ge 0,\\
\infty,& \mbox{ if }x< 0.
\end{cases}
\]
Furthermore, the set of all $\mathcal{H}_2$-exposed points is $[0,\infty)$ and $\overline{\phi}^\ast_{\mathcal{H}_2}$ verifies the condition~\eqref{eq:richExpPoints} for every open set $O\subset\field{R}$. 
Then, (iii) in Theorem \ref{thm:sublinear} gives the LDP
\[
-\underset{y\in {\rm int}(A)}\inf \overline{\phi}^\ast_{\mathcal{H}_2}(y)\le  \underset{n\to\infty}\liminf \tfrac{1}{n}\log\mathbb{P}(X_n\in A)\le
\underset{n\to\infty}\limsup \tfrac{1}{n}\log\mathbb{P}(X_n\in A)\le -\underset{y\in {\rm cl}(A)}\sup \overline{\phi}^\ast_{\mathcal{H}_2}(y),  
\]
where the lower bound
\[
\underset{y\in {\rm int}(A)}\inf \overline{\phi}^\ast_{\mathcal{H}_2}(y)=\inf\{y \colon y \in {\rm int}(A)\cap [0,\infty)\}
\]
is not trivial whenever ${\rm int}(A)\cap [0,\infty)\neq \emptyset$.  
Moreover, we obtain the representation
\[
\overline{\phi}(f)=\underset{n\to\infty}\limsup \frac{1}{n}\log\int_0^\infty e^{n (f(x)-x)}dx=
\underset{x\ge 0}\sup\{f(x)-x\},
\]
for all $f\in C_{\overline{\phi}}(E)$.
\end{example}



\begin{remark}   
The main result of \cite{tan} is a version of G\"{a}rner Ellis theorem for sequences of sublinear expectations on a finite dimensional setting, which is proven by adapting the proof of the standard case. 
This result is a particular instance of Theorem \ref{thm:sublinear} for the special choices  $E=\field{R}^d$ and $\mathcal{H}=(\field{R}^d)^\ast=\field{R}^d$, taking into account (2) in Examples~\ref{examples:linear}. 
As illustrated in Example \ref{ex:classical}, this setting does not cover all $\field{R}^d$ cases in which a LDP exists.    
In addition, the authors of~\cite{tan} consider sublinear expectations of the particular form $\mathcal{E}_n(f)=\sup_{Q\in\mathcal{P}}\mathbb{E}_Q[f(X_n)]$ for some set $\mathcal{P}$ of probabilities measures and a sequence of random variables $(X_n)_{n\in\mathbb{N}}$. 
In particular, this implies that the sublinear $\mathcal{E}_n$ is continuous from below due to the monotone convergence theorem. 
Such a continuity condition is not needed in the present approach. 
Also, among other restrictions it is assumed in \cite{tan} that the limit $\lim_{n\to\infty}\frac{1}{n}\log \mathcal{E}_n(exp(n f))$ exists for all $f\in E^\ast$ (see \cite[Assumption 3.1]{tan}), which was not needed here.      
Furthermore, in \cite{tan} it is not derived the Laplace principle~\eqref{eq:LP2}.
\end{remark}



