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Strong large deviation principles for pair empirical measures of random walks in the Mukherjee-Varadhan topology

Dirk Erhard, Julien Poisat

TL;DR

This work extends the Mukherjee–Varadhan topology to the pair empirical measure $L_n^{(2)}$ of a Markov chain in ${\mathbb R}^d$, proving a strong Large Deviation Principle for its translation-orbit $\widetilde{L}_n^{(2)}$ on a compactified space $\widetilde{\mathcal X}^{(2)}$ with speed $n$ and a rate function given by $\widetilde{J}^{(2)}(\xi)=\sum_{\widetilde{\alpha}\in\xi} h(\alpha\,|\,\alpha_1\otimes \pi)$ when the marginals match, and $+\infty$ otherwise. The rate function encodes deviations in the pair’s marginals and transition structure through relative entropy, and the authors prove its lower semi-continuity, along with matching lower and upper bounds to establish the strong LDP. To transfer results to the single empirical measure, they develop a smoothing projection and a diagonal-tightness framework to perform a contraction despite non-continuity of the naive projection. The paper also adapts the framework to rescaled random walks and demonstrates relevance to translation-invariant problems like the Swiss cheese model for downward deviations of Wiener sausage volume. Collectively, these contributions provide a rigorous tool for analyzing large deviations of empirical measures in non-compact, translation-invariant settings.

Abstract

In this paper we introduce a topology under which the pair empirical measure of a large class of random walks satisfies a strong Large Deviation principle. The definition of the topology is inspired by the recent article by Mukherjee and Varadhan~\cite{MV2016}. This topology is natural for translation-invariant problems such as the downward deviations of the volume of a Wiener sausage or simple random walk, known as the Swiss cheese model~\cite{BBH2001}. We also adapt our result to some rescaled random walks and provide a contraction principle to the single empirical measure despite a lack of continuity from the projection map, using the notion of diagonal tightness.

Strong large deviation principles for pair empirical measures of random walks in the Mukherjee-Varadhan topology

TL;DR

This work extends the Mukherjee–Varadhan topology to the pair empirical measure of a Markov chain in , proving a strong Large Deviation Principle for its translation-orbit on a compactified space with speed and a rate function given by when the marginals match, and otherwise. The rate function encodes deviations in the pair’s marginals and transition structure through relative entropy, and the authors prove its lower semi-continuity, along with matching lower and upper bounds to establish the strong LDP. To transfer results to the single empirical measure, they develop a smoothing projection and a diagonal-tightness framework to perform a contraction despite non-continuity of the naive projection. The paper also adapts the framework to rescaled random walks and demonstrates relevance to translation-invariant problems like the Swiss cheese model for downward deviations of Wiener sausage volume. Collectively, these contributions provide a rigorous tool for analyzing large deviations of empirical measures in non-compact, translation-invariant settings.

Abstract

In this paper we introduce a topology under which the pair empirical measure of a large class of random walks satisfies a strong Large Deviation principle. The definition of the topology is inspired by the recent article by Mukherjee and Varadhan~\cite{MV2016}. This topology is natural for translation-invariant problems such as the downward deviations of the volume of a Wiener sausage or simple random walk, known as the Swiss cheese model~\cite{BBH2001}. We also adapt our result to some rescaled random walks and provide a contraction principle to the single empirical measure despite a lack of continuity from the projection map, using the notion of diagonal tightness.
Paper Structure (14 sections, 25 theorems, 181 equations)

This paper contains 14 sections, 25 theorems, 181 equations.

Table of Contents

  1. Introduction
  2. Topology on the space of probability measures modulo shifts
  3. Vague and weak convergence
  4. Widely separated sequences
  5. Totally disintegrating sequences
  6. Compactification
  7. Large Deviation principles
  8. Lower semi-continuity of the rate function
  9. Lower bound
  10. Upper bound
  11. Adaptation to rescaled random walks
  12. From the pair empirical measure to the empirical measure
  13. Additional observations
  14. On the Mukherjee-Varadhan topology

Key Result

Lemma 2.3

Consider a sequence $(\mu_n)_{n\in \mathbb{N}_0}$ of sub-probability measures in $\mathbb{R}^d$ that converges vaguely to some sub-probability measure $\alpha$. Then we can write $\mu_n= \alpha_n + \beta_n$ for every $n\in\mathbb{N}_0$, where $(\alpha_n)_{n\in\mathbb{N}_0}$ converges weakly to $\alp

Theorems & Definitions (56)

  • Remark 2.1
  • Remark 2.2
  • Lemma 2.3: See Lemma 2.2 in MV2016
  • Remark 2.4
  • Remark 2.5
  • Lemma 2.6
  • proof : Proof of Lemma \ref{['lem:widely-sep']}
  • Lemma 2.7
  • proof : Proof of Lemma \ref{['lem:wid-sep-suff']}
  • Lemma 2.8
  • ...and 46 more