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Sample Path Large Deviations for Random Walks on Regular Trees

Jie Jiang, Shuwen Lai

TL;DR

The paper addresses the problem of sample-path large deviations for nearest-neighbor random walks on $d$-regular trees by focusing on the process $Z_n(t)=\frac{1}{n}l(Y_{[nt]})$ and proving a Mogulskii-type LDP in $L_{\infty}([0,1])$ with a good convex rate function $I$. The rate is characterized implicitly as $I(f)=\sup_{J\in\mathcal{J}} I_J(p_J(f))$, where each finite-dimensional rate $I_J$ is the Fenchel-Legendre transform of the limit log-moment generating function $\Lambda^J$. The proof strategy combines a polygonal approximation $\tilde{Z}_n$, exponential equivalence to $Z_n$, LDPs for finite-dimensional projections, and the Dawson–Gärtner projective-limit theorem to lift results to the path space. For the simple random walk on $T_d$, an explicit rate function is obtained by relating the problem to a biased random walk on $\mathbb{Z}$, enabling a path-space rate $I(f)=\int_0^1 \Lambda^{*}(\dot f(t))\,dt$ on the absolutely continuous functions with $f(0)=0$ and $\|f\|\le1$. The work advances understanding of trajectory-level rare-event probabilities for tree-based random walks and lays groundwork for extensions to other graphs and path-dependent statistics.

Abstract

This paper investigates the large deviation problem in the sample path space of the nearest-neighbor random walks on regular trees. We establish the sample path large deviation principle for the law of the distance from a nearest random walk on a regular tree to the root with a good convex rate function. Furthermore, we derive an implicit expression for the rate function via the Fenchel-Legendre transform of the log-moment generating function.

Sample Path Large Deviations for Random Walks on Regular Trees

TL;DR

The paper addresses the problem of sample-path large deviations for nearest-neighbor random walks on -regular trees by focusing on the process and proving a Mogulskii-type LDP in with a good convex rate function . The rate is characterized implicitly as , where each finite-dimensional rate is the Fenchel-Legendre transform of the limit log-moment generating function . The proof strategy combines a polygonal approximation , exponential equivalence to , LDPs for finite-dimensional projections, and the Dawson–Gärtner projective-limit theorem to lift results to the path space. For the simple random walk on , an explicit rate function is obtained by relating the problem to a biased random walk on , enabling a path-space rate on the absolutely continuous functions with and . The work advances understanding of trajectory-level rare-event probabilities for tree-based random walks and lays groundwork for extensions to other graphs and path-dependent statistics.

Abstract

This paper investigates the large deviation problem in the sample path space of the nearest-neighbor random walks on regular trees. We establish the sample path large deviation principle for the law of the distance from a nearest random walk on a regular tree to the root with a good convex rate function. Furthermore, we derive an implicit expression for the rate function via the Fenchel-Legendre transform of the log-moment generating function.
Paper Structure (15 sections, 15 theorems, 115 equations, 1 figure)

This paper contains 15 sections, 15 theorems, 115 equations, 1 figure.

Table of Contents

  1. Introduction and Main Results
  2. Overview of the proofs
  3. Outline of the article
  4. Preliminaries
  5. Random walks on regular trees
  6. Large deviation and sample path large deviation
  7. Some tools for LDP and sample path LDP
  8. Sample Path Large Deviation for Random Walks
  9. Path Concatenation Method
  10. Large Deviations for Finite-Dimensional Projections
  11. Large Deviations in the Projective Limit Space
  12. Proof of Theorem \ref{['thm-ldp']}
  13. Examples and Questions
  14. Simple Random Walks on Regular Trees
  15. More Questions

Key Result

Theorem 1.1

The measures $(\mu_n )$ in $L_{\infty}([0,1])$ satisfy the LDP with the good rate function where $I_J$ is the rate function of measures $(\mu_n^J)$. Furthermore, $I_J$ is the Fenchel-Legendre transform of $\Lambda^J$ defined by where $\langle \cdot , \cdot \rangle$ is the scalar product in $\mathbb{R}^{j}$. That is

Figures (1)

  • Figure 1: Illustration for the concatenation of paths $\bm{g}_1$ and $\bm{g}_2$.

Theorems & Definitions (28)

  • Theorem 1.1
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Proposition 2.5
  • Proposition 2.6
  • Proposition 2.7
  • Lemma 2.8
  • Lemma 2.9
  • ...and 18 more