Gibbs conditioning principle for log-concave independent random variables

TL;DR

Let $(X_i)$ be independent with log-concave marginals $\nu_i$, and for $\lambda>0$ form $Z^\lambda_i=\sum_x\lambda^x\nu_i(x)$ and $S^\lambda_n$; with $\lambda^*\in(1,\lambda^{\max})$ and $R^*_n=E(S^{\lambda^*}_n)$, the paper proves the Gibbs Conditioning Principle (GCP) that $P((X_i)_{i\le m}\in\cdot|S_n>R^*_n)\to P((X_i^{\lambda^*})_{i\le m}\in\cdot)$ for each fixed $m$, under a condensation-avoidance condition. The proof hinges on two ingredients: stochastic ordering of conditioned tilted measures in $\lambda$ via Efron's theorem on canonical measures, and a convergence lemma showing the conditioning event carries vanishing weight in the limit, enabling a sandwich argument. A condensation-avoidance condition is required to prevent mass transfer to a single site; the results extend Gibbs conditioning from iid/exponential-type settings to non-identically distributed log-concave marginals and connect with canonical-tilted mixtures and queuing system couplings. The work provides a rigorous framework for Gibbs conditioning in a broad non-iid, log-concave context with potential implications for interacting particle systems and related stochastic models.

Abstract

Let $ν_1,ν_2,\dots$ 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 $ν_i$. For $λ>0$ denote $Z^λ_i:= \sum_x λ^xν_i(x)$ and $λ^{\max}:= \sup\{λ>0: Z^λ_i<\infty \text{ for all }i\}$, and assume $λ^{\max}>1$. For $λ<λ^{\max}$, define the tilted probability $ν_i^λ(x):= λ^xν_i(x)/Z^λ_i$, and let $X^λ$ be a sequence of independent variables $X^λ_i$ with law $ν^λ_i$, and denote $S^λ_n:= X^λ_1+\dots+X^λ_n$, with $S_n=S^1_n$. Choose $λ^*\in(1,λ^{\max})$ and denote $R^*_n:= E (S^{λ^*}_n)$. The Gibbs Conditioning Principle (GCP) holds if $P(X\in\cdot|S_n>R^*_n)$ converges weakly to the law of $X^{λ^*}$, as $n\to\infty$. We prove the GCP for log-concave $ν_i$'s, meaning $ν_i(x+1)\,ν_i(x-1) \le ( ν_i(x))^2$, 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 $ν_i$'s, the canonical measures are stochastically ordered with respect to $k$. This, in turn, leads to the ordering of the conditioned tilted measures $P(X^λ\in\cdot|S^λ_n>R^*_n)$ in terms of $λ$. This ordering is a fundamental component of our proof.

Theorems & Definitions (17)

• Definition : Log-concavity
• Theorem 1: Gibbs conditioning principle
• Proposition 2: Stochastic order of conditioned measures
• Lemma 3
• proof
• proof : Proof of Theorem \ref{['thm1']}
• Lemma 4: Order of one-dimensional tilted measures
• Theorem 5: Efron MR171335. Order of canonical measures
• proof : Proof of Proposition \ref{['lem1']}
• Lemma 6