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A threshold for Poisson behavior of non-stationary product measures

Michael Hochman, Nicolò Paviato

TL;DR

The paper identifies a sharp decay threshold for when non-stationary product measures ν, with Bernoulli marginals 1/2+γ_n, yield ν-almost every sequence that is simply Poisson generic. It shows that if γ_n decays faster than log^{-(1/2)} n (up to a δ slack), the Chen–Stein Poisson-approximation framework yields convergence to Poisson(1) in the annealed sense, which implies quenched Poisson genericity. Conversely, slower decay prevents Poisson convergence, even though ν can be singular with respect to the uniform product measure. The results illuminate how singularity and Poisson-pattern randomness can coexist and extend Poisson-genericity insights to broader alphabets and non-stationary settings.

Abstract

Let $γ_{n}= O (\log^{-c}n)$ and let $ν$ be the infinite product measure whose $n$-th marginal is Bernoulli$(1/2+γ_{n})$. We show that $c=1/2$ is the threshold, above which $ν$-almost every point is simply Poisson generic in the sense of Peres-Weiss, and below which this can fail. This provides a range in which $ν$ is singular with respect to the uniform product measure, but $ν$-almost every point is simply Poisson generic.

A threshold for Poisson behavior of non-stationary product measures

TL;DR

The paper identifies a sharp decay threshold for when non-stationary product measures ν, with Bernoulli marginals 1/2+γ_n, yield ν-almost every sequence that is simply Poisson generic. It shows that if γ_n decays faster than log^{-(1/2)} n (up to a δ slack), the Chen–Stein Poisson-approximation framework yields convergence to Poisson(1) in the annealed sense, which implies quenched Poisson genericity. Conversely, slower decay prevents Poisson convergence, even though ν can be singular with respect to the uniform product measure. The results illuminate how singularity and Poisson-pattern randomness can coexist and extend Poisson-genericity insights to broader alphabets and non-stationary settings.

Abstract

Let and let be the infinite product measure whose -th marginal is Bernoulli. We show that is the threshold, above which -almost every point is simply Poisson generic in the sense of Peres-Weiss, and below which this can fail. This provides a range in which is singular with respect to the uniform product measure, but -almost every point is simply Poisson generic.
Paper Structure (8 sections, 9 theorems, 54 equations)

This paper contains 8 sections, 9 theorems, 54 equations.

Table of Contents

  1. Introduction
  2. Setup and notation
  3. Convergence to Poisson
  4. A general convergence theorem applied to our setting
  5. $A_{k}\rightarrow0$
  6. $B_{k}\rightarrow0$
  7. $C_{k}\to0$
  8. Non-convergence

Key Result

Theorem 1.1

Suppose that $\gamma_n\in(-1/2,1/2)$ and $\nu$ is the corresponding product measure. If $\gamma_n= O (\log^{-(1/2+\delta)}n)$ for some $\delta>0$, then $\nu$-almost every $x\in\Omega^{\mathbb{N}}$ is simply Poisson generic. On the other hand, if $\gamma_n=\log^{-(1/2-\delta)}n$ for all large $n$, t

Theorems & Definitions (18)

  • Theorem 1.1
  • Remark 1.2
  • Proposition 3.1
  • Proposition 3.2
  • proof
  • Theorem 3.3
  • Lemma 3.4
  • proof
  • Remark 3.5
  • Proposition 3.6
  • ...and 8 more