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Polynomial Tail Decay for Stationary Measures

Samuel Kittle, Constantin Kogler

TL;DR

The paper analyzes polynomial tail decay for stationary measures arising from contracting-on-average random walks on complete metric spaces. It proves existence and uniqueness of a stationary measure $\nu$ for finitely supported $\mu$ with contraction rate $\chi_{\mu}<0$ and establishes a polynomial tail bound $\nu(\{d(x,y) \ge R\}) \ll R^{-\alpha}$, with a Lyapunov-exponent variant $\lambda_{\mu}<0$ under a large-deviation hypothesis. The approach relies on constructing $\nu$ as a limit of random products and applying a large-deviation principle, avoiding renewal theory and extending to non-finitely supported $\mu$ via the compact-open topology. The work also discusses consequences for self-similar and self-affine measures, including a polynomial lower bound in the self-similar setting and finite differential entropy for polynomial-tail stationary measures, connecting to broader results in the literature.

Abstract

We show on complete metric spaces a polynomial tail decay for stationary measures of contracting on average generating measures.

Polynomial Tail Decay for Stationary Measures

TL;DR

The paper analyzes polynomial tail decay for stationary measures arising from contracting-on-average random walks on complete metric spaces. It proves existence and uniqueness of a stationary measure for finitely supported with contraction rate and establishes a polynomial tail bound , with a Lyapunov-exponent variant under a large-deviation hypothesis. The approach relies on constructing as a limit of random products and applying a large-deviation principle, avoiding renewal theory and extending to non-finitely supported via the compact-open topology. The work also discusses consequences for self-similar and self-affine measures, including a polynomial lower bound in the self-similar setting and finite differential entropy for polynomial-tail stationary measures, connecting to broader results in the literature.

Abstract

We show on complete metric spaces a polynomial tail decay for stationary measures of contracting on average generating measures.

Paper Structure

This paper contains 6 sections, 7 theorems, 34 equations.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['HutchinsonSuperGen1']} and Theorem \ref{['HutchinsonLyapunovGen']}
  3. Further Remarks
  4. Finiteness of Differential Entropy
  5. A generalisation of Theorem \ref{['HutchinsonSuperGen1']}
  6. Polynomial Lower Bound for Self-Similar Measures

Key Result

Theorem 1.1

Let $X$ be a complete metric space and let $\mu$ be a finitely supported probability measure on $L(X)$ with $\chi_{\mu} < 0$. Then there exists a unique probability measure $\nu$ on $X$ such that $\mu * \nu = \nu$. Moreover, there is $\alpha = \alpha(\mu) > 0$ such that for all $x \in X$ and $R > 0$ where the implied constant depends on $\mu$ and $x$.

Theorems & Definitions (13)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • proof
  • proof
  • Lemma 3.1
  • proof
  • ...and 3 more