A threshold for Poisson behavior of non-stationary product measures

TL;DR

This paper identifies a critical threshold for when non-stationary product measures exhibit Poisson behavior, revealing a phase transition at a specific decay rate of the Bernoulli parameters.

Contribution

It establishes that the decay rate c=1/2 is the threshold for Poisson genericity in non-stationary product measures, connecting measure singularity and Poisson behavior.

Findings

Threshold c=1/2 determines Poisson genericity

Measures with c>1/2 are Poisson generic almost surely

Measures with c<1/2 can fail to be Poisson generic

Abstract

Let $\gamma_{n}= O (\log^{-c}n)$ and let $\nu$ be the infinite product measure whose $n$-th marginal is Bernoulli$(1/2+\gamma_{n})$. We show that $c=1/2$ is the threshold, above which $\nu$-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 $\nu$ is singular with respect to the uniform product measure, but $\nu$-almost every point is simply Poisson generic.