Lossy Asymptotic Equipartition Property For Geometric Networked Data Structures

TL;DR

This paper extends the Asymptotic Equipartition Property to geometric networked data, specifically wireless sensor networks modeled as colored geometric random graphs, using large deviation principles.

Contribution

It introduces a generalized AEP for CGRGs and applies it to real sensor network data for water quality monitoring.

Findings

Extended AEP to CGRGs using large deviation techniques

Validated theoretical results with real sensor network data

Provided insights into data compression for geometric network models

Abstract

This article extends the Generalized Asypmtotic Equipartition Property of Networked Data Structures to cover the Wireless Sensor Network modelled as coloured geometric random graph (CGRG). The main techniques used to prove this result remains large deviation principles for properly defined empirical measures on CGRGs. As a motivation for this article, we apply our results to some data from Wireless Sensor Network for Monitoring Water Quality from a Lake..

Equations

$${\mathcal{L}}_{n,[z]}(\beta_{x}(z),\beta_{y}(z))=\frac{1}{n}\sum_{v\in[n]}\delta_{\big{(}{\mathcal{B}}_{X}(z_{v}),\,{\mathcal{B}}_{Y}(z_{v})\big{)}}(\beta_{x}(z),\beta_{y}(z)),\,\mbox{ for $(\beta_{x}(z),\beta_{y}(z))\in{\mathcal{M}}[{({\mathcal{X}}\times{\mathcal{N}}({\mathcal{X}}))}^{2}].$ }$$

$${\mathcal{L}}_{n,[z],1}(\beta_{x}(z)):=\frac{1}{n}\sum_{v\in[n]}\delta_{\big{(}{\mathcal{B}}_{X}(z_{v})\big{)}}(\beta_{x}(z))\,\mbox{and}\,{\mathcal{L}}_{n,[z],2}(\beta_{y}(z)):=\frac{1}{n}\sum_{v\in[n]}\delta_{\big{(}{\mathcal{B}}_{Y}(z_{v})\big{)}}(\beta_{y}(z))\,$$

$$\sigma^{(n)}(x,y)=\frac{1}{n}\sum_{i\in[n]}\sigma\Big{(}{\mathcal{B}}_{x}(z_{i}),\,{\mathcal{B}}_{y}(z_{i})\Big{)},$$

$$B(x,\alpha)=\Big{\{}y\in\hat{G}_{[z]}:\,\,\sigma^{(n)}(x,y)\leq\alpha\Big{\}}.$$

$$p_{\pi\omega}(a,l)=\pi(a)\prod_{b\in{\mathcal{X}}}\frac{e^{-\omega(a,b)/\pi(a)}[\omega(a,b)/\pi(a)]^{\ell(b)}}{\ell(b)!},\,\mbox{for $\ell\in{\mathcal{N}}({\mathcal{X}})$ }$$

$$\displaystyle J_{1}(\mu)=\left\{\begin{array}[]{ll}H\big{(}\mu\,\|\,p_{\pi\omega}\otimes p_{\pi\omega}),&\mbox{if $\mu$ is consistent and $\mu_{1,1}=\mu_{1,2}=\pi$,}\\ \infty&\mbox{otherwise,}\par\end{array}\right.$$

$$\,p_{\pi\omega}\otimes p_{\pi\omega}\big{(}(a_{x},a_{y}),(l_{x},l_{y})\big{)}=p_{\pi\omega}(a_{x},\,l_{x})p_{\pi\omega}(a_{y},\,l_{y}).$$

$${\alpha }_{av}(\pi ,\omega )=⟨\log ⟨e^{t\sigma ({\mathcal{B}}_X,{\mathcal{B}}_Y)},p_{\pi \omega }⟩,p_{\pi \omega }⟩.$$

$$\alpha_{min}^{(n)}(\pi,\omega)=\mathbb{E}_{P_{x}^{(n)}}\big{[}{\rm essinf}\,_{Y\,\approx\,Q_{y}^{(n)}}\sigma^{(n)}(X,Y)\big{]}\,\mbox{ $\to\alpha_{min}(\pi,\omega).$}$$

$$R_{n}(P_{n}^{(x)},Q_{n}^{(y)},\alpha):=\inf_{V_{n}}\Big{\{}\frac{1}{n}H(V_{n}\,\|\,P_{n}^{(x)}\times Q_{n}^{(y)}):\,V_{n}\in{\mathcal{M}}({\mathcal{G}}\times\hat{{\mathcal{G}}})\Big{\}}$$

$$\alpha_{min}^{\infty}(\pi,\omega):=\inf\Big{\{}\alpha\geq 0:\,\sup_{n\geq 1}R_{n}(P_{x}^{(n)},Q_{y}^{(n)},\alpha)<\infty\Big{\}}.$$

$$J_{\sigma}(t):=\inf_{\mu}\Big{\{}J_{1}(\mu):\,\langle\sigma,\,\mu\rangle=t\Big{\}}.$$

$$\lim_{n\to\infty}-\frac{1}{n}\log Q_{x}^{(n)}\Big{(}B(X^{[z]},\alpha)\Big{)}=R\big{(}\mathbb{P}_{x}^{(\pi\omega)},\mathbb{P}_{y}^{(\pi\omega)},\alpha\big{)}\,\,\mbox{almost surely,}$$

$$\displaystyle R(P,Q,\alpha)=\left\{\begin{array}[]{ll}0,&\mbox{ if \,$\alpha\geq 2\omega^{\Delta(d)}(SG,\,SI)+\omega^{\Delta(d)}(SG,\,SG)+\omega^{\Delta(d)}(SI,\,SI)+2\omega^{\Delta(d)}(SI,\,SG)$.}\\ \infty&\mbox{otherwise,}\par\end{array}\right.$$

$${\mathcal{L}}_{n}(\beta_{x},\beta_{y})=\frac{1}{n}\sum_{v\in[n]}\delta_{\big{(}{\mathcal{B}}_{X}(v),\,{\mathcal{B}}_{Y}(v)\big{)}}(\beta_{x},\beta_{y}),\,\mbox{ for $(\beta_{x},\beta_{y})\in{\mathcal{M}}[{({\mathcal{X}}\times{\mathcal{N}}({\mathcal{X}}))}^{2}].$ }$$

$$\displaystyle{\mathcal{C}}_{\pi\omega}^{\varepsilon}=\Big{\{}\mu\in{\mathcal{M}}\colon\sup_{\beta_{x},\beta_{y}\in{\mathcal{X}}\times{\mathcal{N}}({\mathcal{X}})}|\mu(\beta_{x},\,\beta_{y})-p_{\pi\omega}\otimes p_{\pi\omega}(\beta_{x},\,\beta_{y})|\geq\varepsilon\Big{\}}.$$

$$\limsup_{n\to\infty}\frac{1}{n}\log\mathbb{P}_{(\pi_{n},\omega_{n})}\big{(}{\mathcal{C}}_{\pi\omega}^{\varepsilon}\big{)}\leq-\inf_{\mu\in{\mathcal{C}}^{\varepsilon}}J_{1}(\mu).$$

$$\Lambda_{\sigma}:=\big{\{}\mu:\langle\sigma,\,\mu\rangle\in\Lambda\big{\}}$$

$$-\underset{t\in In(\Lambda )}{\inf }{J}_{\sigma }(t)$$

$$\displaystyle\leq\liminf_{n\to\infty}\mbox{$\frac{1}{n}$}\log\mathbb{P}\Big{\{}\sigma^{(n)}(X^{[z]},Y^{[z]})\in\Lambda\big{|}X^{[z]}=x,\,\Psi({\mathcal{L}}_{n,[z],1})=\Psi({\mathcal{L}}_{n,[z],2})=(\pi_{n},\omega_{n})\Big{\}}$$

$$\displaystyle\leq\lim_{n\to\infty}\mbox{$\frac{1}{n}$}\log\mathbb{P}\Big{\{}\sigma^{(n)}(X^{[z]},Y^{[z]})\in\Lambda\big{|}X^{[z]}=x,\,\Psi({\mathcal{L}}_{n,[z],1})=\Psi({\mathcal{L}}_{n,[z],2})=(\pi_{n},\omega_{n})\Big{\}}$$

$$\displaystyle\leq\limsup_{n\to\infty}\mbox{$\frac{1}{n}$}\log\mathbb{P}\Big{\{}\sigma^{(n)}(X^{[z]},Y^{[z]})\in\Lambda\big{|}X^{[z]}=x,\,\Psi({\mathcal{L}}_{n,[z],1})=\Psi({\mathcal{L}}_{n,[z],2})=(\pi_{n},\omega_{n})\Big{\}}$$

$$\le -\underset{\mu \in cl({\Lambda }_{\sigma })}{\inf }{J}_1(\mu )=-\underset{t\in cl(\Lambda )}{\inf }{J}_{\sigma }(t).$$

$$R({\mathbb{P}}_x^{\pi \omega },{\mathbb{P}}_y^{\pi \omega },\alpha )=\underset{t\in \mathbb{R}}{\sup }[t\alpha -{\mathcal{H}}_{\infty }(t)]={\mathcal{H}}_{\infty }^{\ast }(\alpha )$$

$${\mathcal{H}}_{\infty}^{*}(t):=\lim_{n\to\infty}\mbox{$\frac{1}{n}$}\log\int e^{nt\Big{\langle}\sigma,\,{\mathcal{L}}_{n,[z]}\Big{\rangle}}Q_{y}^{(n)}(dy)$$

$${\mathcal{H}}_{\infty}(t):=\lim_{n\to\infty}{\mathcal{H}}_{n}(t)=\lim_{n\to\infty}\mbox{$\frac{1}{n}$}\int\Big{[}\log\int e^{nt\Big{\langle}\sigma,\,{\mathcal{L}}_{n,[z]}\Big{\rangle}}Q_{y}^{(n)}(dy)\Big{]}P_{x}^{(n)}(dx).$$

$$\mbox{$\frac{1}{n}$}{\mathcal{H}}_{n}(nt)=\frac{1}{n}\sum_{j=1}^{n}\log\mathbb{E}_{Q_{y}^{(n)}}\big{(}e^{t\sigma({\mathcal{B}}_{x}(j),{\mathcal{B}}_{y}(j)}\big{)}\to\langle\log\langle e^{t\sigma({\mathcal{B}}_{X^{[z]}},{\mathcal{B}}_{Y^{[z]}})},p_{\pi\omega}\rangle,p_{\pi\omega}\rangle=\alpha_{av}(\pi,\omega).$$

$$\alpha_{min}^{(n)}(\pi,\omega):=\lim_{t\downarrow-\infty}\mbox{$\frac{{\mathcal{H}}_{n}(t)}{t}$}$$

$$R_{n}(P_{x}^{(n)},Q_{y}^{(n)},\alpha)=\sup_{t\in\mathbb{R}}\big{(}t\alpha-{\mathcal{H}}_{n}(t)\big{)}:={\mathcal{H}}_{n}^{*}(\alpha)$$

$${\mathcal{H}}_n^{\ast }(\alpha )=\underset{\delta \to 0}{\lim }\underset{n\to \infty }{\mathrm{limsup}}\underset{|\hat{\alpha }-\alpha |<\delta }{\inf }{\mathcal{H}}_n^{\ast }(\hat{\alpha }).$$

Full text

Lossy Asymptotic Equipartition Property for Geometric Networked Data Structures

By Kwabena Doku-Amponsah

††Mathematics Subject Classification : 94A15, 94A24, 60F10, 05C80††*Keywords: * Information theory, rate-distortion theory,exponential equivalent measures, relative entropy, geometric networked data structures, wireless sensor networks.††Address: Statistics Department, University of Ghana, Box LG 115, Legon,Ghana.  E-mail: kdoku@ug.edu.gh.

University of Ghana

Abstract. This article extends the Generalized Asypmtotic Equipartition Property of Networked Data Structures to cover the Wireless Sensor Network modelled as coloured geometric random graph (CGRG). The main techniques used to prove this result remains large deviation principles for properly defined empirical measures on CGRGs. As a motivation for this article, we apply our results to some data from Wireless Sensor Network for Monitoring Water Quality from a Lake.

1. Introduction

Field data we often encounter from the study of the environment are usually structured according to geometry and the connectivity between the locations that make up the environment. Example,data from (i) monitoring air quality at key industrial sites, (ii) looking for key contaminating agents from the exhausts of public buses, (iii) monitoring the cleanliness in lakes and many more, are all structured according to the geometry of the area of study and the connectivity of the location that make up the environment. To design and implement simplex (Linear programming) algorithm for the solution of generalized network flow problems of the geometric structured network data ,see example [1], or to find an efficient coding scheme or an approximate pattern matching algorithms, see example [2], we need an information theory for such data structures, and the lossy Asymptotic Equipartition Property (AEP) for the geometric networked data structures is key to finding an information theory for the data structure. See [6] and [7] for similar results for other types of data structures.

The aim of this article is to extend the Lossy AEP for Networked Data Structures modelled as Coloured Random Graph (CRG), see [6, Theorem 2.1], to cover the WSN. To be specific we model the Geometric Networked Data Structures ( WSN) as a CGRG and use some of the large deviation techniques developed in [7] to prove a strong law of large numbers (SLLN), see Lemma 3.3, for the random network. Using the SLLN and the techniques deployed in [9] we extend the Lossy AEP to cover the WSN.

The remaining part of the paper is organized as follows: Section 2 contains the main result of the paper and an application to some data from environmental science. See, Theorem 2.1 in Subsection 2.1 and the application in Subsection 2.2. Section 3 gives the proof of the main result; starting with the LDPs (Lemmas 3.1 and 3.2) in Subsection 3.1, followed by statement and proof of a strong law of large numbers, see Lemma 3.3 and ending with derivation of the main results from the SLLN in subsection 3.2.

2. Generalized AEP for CGRG Process

2.1. Main Result

We consider two CGRG processes X^{[z]}=\big{\{}(X(z_{1}),X(z_{2})):\,z_{i}\,z_{j}\in E,\,i,j=1,2,3,...,n,i\not=j\big{\}} and Y^{[z]}=\big{\{}(Y(z_{i}),Y(z_{j})):\,z_{i}\,z_{j}\in E,\,i,j=1,2,3,...,n,i\not=j\big{\}} which take values in $G_{[z]}=G({\mathcal{X}},\,z_{1},z_{2},z_{3},...,z_{n})$ and $\hat{G}_{[z]}=\hat{G}({\mathcal{X}},\,z_{1},z_{2},z_{3},...,z_{n}),$ resp., the spaces of finite graphs on ${\mathcal{X}}$ and $\,z_{1},z_{2},z_{3},...,z_{n}\in[0,\,1]^{d}.$ We equip $G_{[z]}$, $\hat{G}_{[z]}$ with their Borel $\sigma-$fields ${\mathcal{F}}_{x}$ and $\hat{{\mathcal{F}}}_{x}.$ Let $\mathbb{P}_{x}$ and $\mathbb{P}_{y}$ denote the probability measures of the entire processes $X^{[z]}$ and $Y^{[z]}.$ By $\mathbb{P}_{x}^{(\pi\omega)}$ and $\mathbb{P}_{y}^{(\pi\omega)}$ we denote the coloured geometric random graphs $X^{[z]}$ and $Y^{[z]}$ conditioned to have empirical colour measure $\pi$ and empirical pair measure $\omega.$ See, example [3]. We always assume that $X^{[z]}$ and $Y^{[z]}$ are independent of each other.

By ${\mathcal{X}}$ we denote a finite alphabet and denote by ${\mathcal{N}}({\mathcal{X}})$ the space of counting measure on ${\mathcal{X}}$ equipped with the discrete topology. By ${\mathcal{M}}({\mathcal{X}})$ we denote the space of probability measures on ${\mathcal{X}}$ equipped with the weak topology and ${\mathcal{M}}_{*}({\mathcal{X}})$ the space of finite measures on ${\mathcal{X}}$ equipped with the weak topology.

We define the process-level empirical measure ${\mathcal{L}}_{n,[z]}$ induced by $X^{[z]}$ and $Y^{[z]}$ on $G_{[z]}\times\hat{G}_{[z]}$ by

[TABLE]

[TABLE]

for $(\beta_{x}(z),\beta_{y}(z))\in{\mathcal{M}}[{({\mathcal{X}}\times{\mathcal{N}}({\mathcal{X}}))}^{2}].$

Throughout the rest of the article we will assume that $X^{[z]}$ and $Y^{[z]}$ are CGRG processes, See [11]. For $n\geq 1$, let $P_{x}^{(n)}$ denote the marginal distribution of $X^{[z]}$ on $[n]=\{1,2,3,...,n\}$ taking with respect to $\mathbb{P}_{x}^{(\pi\omega)}$ and $Q_{y}^{(n)}$ denote the marginal distribution $Y^{[z]}$ on $[n]=\{1,2,3,...,n\}$ with respect to $\mathbb{P}_{y}^{(\pi\omega)}.$

Let $\sigma:{\mathcal{X}}\times{\mathcal{N}}({\mathcal{X}})\times{\mathcal{X}}\times{\mathcal{N}}({\mathcal{X}})\to[0,\infty)$ be an arbitrary non-negative function and define a sequence of single-letter distortion measures $\sigma^{(n)}:G_{[z]}\times\hat{G}_{[z]}\to[0,\infty),$ $n\geq 1$ by

[TABLE]

where ${\mathcal{B}}_{x}(z_{i})=(x(z_{i}),L_{x}(z_{i}))$ and ${\mathcal{B}}_{y}(z_{i})=(y(z_{i}),L_{y}(z_{i})).$ Given $\alpha\geq 0$ and $x\in G_{[z]}$ , we denote the distortion-ball of radius $\alpha$ by

[TABLE]

We shall call the measure $\mu\in{\mathcal{M}}[{({\mathcal{X}}\times{\mathcal{N}}({\mathcal{X}}))}^{2}]$ consistent if $\mu_{1}$, $\mu_{2}$ are both consistent marginals of $\mu.$ Refer to [7, Equation 2.1] for the concept of consistent measures.

For $(\pi,\,\omega)\in{\mathcal{M}}({\mathcal{X}})\times{\mathcal{M}}({\mathcal{X}}\times{\mathcal{X}}),$ we write

[TABLE]

and define the rate function $I_{1}:{\mathcal{M}}[{({\mathcal{X}}\times{\mathcal{N}}({\mathcal{X}}))}^{2}]\to[0,\,\infty]$ by

[TABLE]

where

[TABLE]

By $x\,\approx\,p$ we mean $x$ has distribution $p.$ For $(\pi,\,\omega)\in{\mathcal{M}}({\mathcal{X}})\times{\mathcal{M}}({\mathcal{X}}\times{\mathcal{X}}),$ we write

[TABLE]

Assume

[TABLE]

For $n>1,$ we write

[TABLE]

and

[TABLE]

Theorem 2.1 (ii) below provides a Lossy AEP for WSN data structures.

Theorem 2.1**.**

Suppose $X^{[z]}$ and $Y^{[z]}$ are CGRG process. Assume $\sigma$ are bounded function. Then,

• (i)

with $\mathbb{P}_{(x)}^{(\pi\omega)}-$ probability $1,$ conditional on the event \big{\{}\,\Psi({\mathcal{L}}_{n,[z],1})=\Psi({\mathcal{L}}_{n,[z],2})=(\pi,\omega)\big{\}} the random variables \Big{\{}\sigma^{(n)}(x,Y^{[z]})\Big{\}} satisfy an LDP with deterministic, convex rate-function

[TABLE]

• (ii)

for all \alpha\in\Big{(}\alpha_{min}(\pi,\omega),\,\alpha_{av}(\pi,\omega)\Big{)}, except possibly at $\alpha=\alpha_{min}^{\infty}(\pi,\omega)$

[TABLE]

where $R(p,q,\alpha)=\inf_{\mu}H(\mu\,\|\,p\times q).$

2.2. Application:

Wireless Sensor Network for Monitoring Water Quality from a Lake. Let consider a WSN (to monitor the cleanliness in lakes, particularly those used as sources of drinking water) consisting of sensors capable of carrying out some processing, gathering sensory information and communicating with other connected nodes in the network modelled as coloured geometric random graph on $n$ location, say $z_{1},z_{2},...,z_{n}.$ By $SG$ we denote sensors capable of carrying out some processing, gathering sensory information while communicating with other sensors and $SI$ sensors gathering sensory information while communicating with other sensors. Suppose the locations are $z_{1},z_{2},...,z_{n}\in[0,\,1]^{d}$ partition into $n\pi_{n}(SG)$ block of $SG$ and $n\pi_{n}(SI)$ block of $SI,$ and $n\|\omega_{n}^{\Delta(d)}\|$ number of communication links divided into $n\omega_{n}^{\Delta(d)}(SG,\,SI),$ $n\omega_{n}^{\Delta(d)}(SI,\,SG),$ $n\omega_{n}^{\Delta(d)}(SG,\,SG)/2,$ $n\omega_{n}^{\Delta(d)}(SI,\,SI)/2$ different interactions, respectively, for $\Delta(d)$ a function which depends on the connectivity radius of the WSN. Assume $\pi_{n}$ converges $\pi$ and $\omega_{n}^{\Delta(d)}$ converges $\omega^{\Delta(d)}.$ If we take $\sigma(s,r)=(s-r)^{2}$ then, by Theorem 2.1 we have the rate-distortion of

[TABLE]

where \omega^{\Delta(d)}(a,b)=\mbox{\frac{\pi^{d/2}}{\big{[}d/2\big{]}!}}\lambda_{[d]}(a,b))\pi(a)\pi(b). See, [6] for the relationship between the connectivity radius and $\lambda_{[d]}.$ We refer to [13] for more on modelling of the physical environment using the Wireless Sensor Network.

3. Proof of Theorem 2.1.

3.1. LDPs.

Recall from [7] that X=\big{\{}(X(u),X(v)):\,uv\in E\big{\}} and Y=\big{\{}(Y(u),Y(v)):\,uv\in E\big{\}} are CRG processes with values from $G=G({\mathcal{X}})$ and $\hat{G}=\hat{G}({\mathcal{X}}),$ resp., the spaces of finite graphs on ${\mathcal{X}}.$

We define the process-level empirical measure ${\mathcal{L}}_{n}$ induced by $X$ and $Y$ on $G\times\hat{G}$ by

[TABLE]

Lemma 3.1** (Exponential Equivalence).**

Suppose $(X^{[z]},Y^{[z]})$ are CGRG on the $d-$ dimensional Torus and $(X,Y)$ are CRG. Then, conditional on the event \big{\{}\,\Psi({\mathcal{L}}_{n,^{[}z],1})=\Psi({\mathcal{L}}_{n,[z],2})=\Psi({\mathcal{L}}_{n,1})=\Psi({\mathcal{L}}_{n,2})=(\pi,\omega)\big{\}} the law of ${\mathcal{L}}_{n,[z]}$ is exponentially equivalent to the law of ${\mathcal{L}}_{n}^{.}$

Proof.

We denote by $(\tilde{X},\tilde{Y})$ the random allocation process and notice from [6, Lemma 3.1] and [5, Lemma 0.4] that conditional on \big{\{}\,\Psi({\mathcal{L}}_{n,[z],1})=\Psi({\mathcal{L}}_{n,[z],2})=\Psi({\mathcal{L}}_{n,1})=\Psi({\mathcal{L}}_{n,2})=(\pi,\omega)\big{\}} the law of $(X^{[z]},Y^{[z]})$ is exponentially equivalent to the law of $(\tilde{X},\tilde{Y})$ and the law of $(\tilde{X},\tilde{Y})$ is exponentially equivalent to the law of $(X,Y).$ Therefore, conditional on \big{\{}\,\Psi({\mathcal{L}}_{n,[z],1})=\Psi({\mathcal{L}}_{n,[z],2})=\Psi({\mathcal{L}}_{n,1})=\Psi({\mathcal{L}}_{n,2})=(\pi,\omega)\big{\}} we have $(X^{[z]},Y^{[z]})$ exponentially equivalent to $(X,Y).$

∎

Lemma 3.2** (LDP).**

Suppose $(X^{[z]},Y^{[z]})$ are coloured geometric random graph on the $d-$ dimensional Torus. Then, conditional on the event \big{\{}\,\Psi({\mathcal{L}}_{n,[z],1})=\Psi({\mathcal{L}}_{n,[z],2})=(\pi,\omega)\big{\}} the law of ${\mathcal{L}}_{n,[z]}$ obeys a process level LDP with good rate function $J_{1}$

The proof of this Lemma 3.2 which follows from 3.1 [7, Theorem ] and [10, Theorem 4.2.13], is omitted from the paper.

3.2. Derivation of the AEP.

We write ${\mathcal{M}}:={\mathcal{M}}[({\mathcal{X}}\times{\mathcal{N}}({\mathcal{X}}))^{2}]$ and define the set ${\mathcal{C}}^{\varepsilon}$ by

[TABLE]

Lemma 3.3** (SLLN).**

Suppose the sequence of measures $(\pi_{n},\omega_{n})$ converges to the pair of measures $(\pi_{n},\omega_{n}).$ For any $\varepsilon>0$ we have \lim_{n\to\infty}\mathbb{P}_{(\pi_{n},\omega_{n})}\big{(}{\mathcal{C}}_{\pi\omega}^{\varepsilon}\big{)}=0.

Observe that ${\mathcal{C}}_{\pi\omega}^{\varepsilon}$ defined above is a closed subset of ${\mathcal{M}}$ and so by Lemma 3.2 we have that

[TABLE]

We use proof by contradiction to show that the right hand side of (3.1) is negative.Suppose that there exists sequence $\mu_{n}$ in ${\mathcal{C}}_{\pi\omega}^{\varepsilon}$ such that $J_{1}(\mu_{n})\downarrow 0.$ Then, there is a limit point $\mu\in F_{1}$ with $J_{1}(\mu)=0.$ Note $J_{1}$ is a good rate function and its level sets are compact, and the mapping $\mu\mapsto J_{1}(\mu)$) lower semi-continuity. Now $J_{1}(\mu)=0$ implies $\mu(\beta_{x},\,\beta_{y})=p_{\pi\omega}\otimes p_{\pi\omega}(\beta_{x},\,\beta_{y}),$ for all $\beta_{x},\beta_{y}\in{\mathcal{X}}\times{\mathcal{N}}({\mathcal{X}})$ which contradicts $\mu\in{\mathcal{C}}_{\pi\omega}^{\varepsilon}$.

(i) Notice $\displaystyle\sigma^{(n)}(X^{[z]},Y^{[z]})=\langle\sigma,\,{\mathcal{L}}_{n,[z]}\rangle$ and if $\Lambda$ is open (closed) subset of ${\mathcal{M}}$ then

[TABLE]

is also open (closed) set since $\sigma$ is bounded function.

[TABLE]

(ii) Observe that $\sigma$ are bounded, therefore by Varadhan’s Lemma and convex duality, we have

[TABLE]

where

[TABLE]

exits for $\mathbb{P}$ almost everywhere $x.$ Using bounded convergence, we can show that

[TABLE]

Using Lemma 3.3, by boundedness of $\sigma$ we have that

[TABLE]

Also let

[TABLE]

so that ${\mathcal{H}}_{n}^{*}(\alpha)=\infty$ for $\alpha<\alpha_{min}^{(n)}(\pi,\omega)$, while ${\mathcal{H}}_{n}^{*}(\alpha)<\infty$ for $\alpha>\alpha_{min}^{(n)}(\pi,\omega).$ Observe that for $n<\infty$ we have \alpha_{min}^{(n)}(\pi,\omega)=\mathbb{E}_{P_{x}^{(n)}}\big{[}{\rm essinf}\,_{Y\,\approx\,Q_{y}^{(n)}}\sigma^{(n)}(X^{[z]},Y^{[z]})\big{]}, which converges to $\alpha_{min}(\pi,\omega).$ Applying similar arguments as [9, Proposition 2] we obtain

[TABLE]

Now we observe from [9, Page 41] that the converge of ${\mathcal{H}}_{n}^{*}(\cdot)\to{\mathcal{H}}_{\infty}(\cdot)$ is uniform on compact subsets of $\mathbb{R}.$ Moreover, ${\mathcal{H}}_{n}$ is convex, continuous functions converging informally to ${\mathcal{H}}_{\infty}$ and hence we can invoke [12, Theorem 5] to obtain

[TABLE]

Applying similar arguments as [9, Page 41] in the lines after equation (64) we have (2.3) which completes the proof.

Conflict of Interest

The author declares that he has no conflict of interest.

Acknowledgement

This extension has been mentioned in the author’s PhD Thesis at University of Bath.