Gibbs conditioning principle for log-concave independent random variables

Abstract

Let $\hat{I}{½}_1,\hat{I}{½}_2,\dots$ be a sequence of probabilities on the nonnegative integers, and $X=(X_1,X_2,\dots )$ be a sequence of independent random variables $X_i$ with law $\hat{I}{½}_i$. For $\hat{I}»>0$ denote $Z^{\hat{I}}{»}_i:={\sum }_x\hat{I}{»}^x\hat{I}{½}_i(x)$ and $\hat{I}{»}^{\max }:=\sup \{\hat{I}»>0:Z^{\hat{I}}{»}_i<\infty \text{ for all }i\}$, and assume $\hat{I}{»}^{\max }>1$. For $\hat{I}»<\hat{I}{»}^{\max }$, define the tilted probability $\hat{I}{½}_i^{\hat{I}}»(x):=\hat{I}{»}^x\hat{I}{½}_i(x)/Z^{\hat{I}}{»}_i$, and let $X^{\hat{I}}»$ be a sequence of independent variables $X^{\hat{I}}{»}_i$ with law $\hat{I}{½}^{\hat{I}}{»}_i$, and denote $S^{\hat{I}}{»}_n:=X^{\hat{I}}{»}_1+\cdots +X^{\hat{I}}{»}_n$, with $S_n=S_n^1$. Choose $\hat{I}{»}^{\ast }\in (1,\hat{I}{»}^{\max })$ and denote $R_n^{\ast }:=E(S_n^{\hat{I}{»}^{\ast }})$. The Gibbs Conditioning Principle (GCP) holds if $P(X\in \cdot |S_n>R_n^{\ast })$ converges weakly to the law of $X^{\hat{I}{»}^{\ast }}$, as $n\to \infty$. We prove the GCP for log-concave $\hat{I}{½}_i$'s, meaning $\hat{I}{½}_i(x+1)\hat{I}{½}_i(x-1)\le (\hat{I}{½}_i(x){)}^2$, subject to a technical condition that prevents condensation. The canonical measures are the distributions of the first $n$ variables, conditioned on their sum being $k$. Efron's theorem states that for log-concave $\hat{I}{½}_i$'s, the canonical measures are stochastically ordered with respect to $k$. This, in turn, leads to the ordering of the conditioned tilted measures $P(X^{\hat{I}}»\in \cdot |S^{\hat{I}}{»}_n>R_n^{\ast })$ in terms of $\hat{I}»$. This ordering is a fundamental component of our proof.

14 pages