Gibbs conditioning principle for log-concave independent random variables
Let $\nu_1,\nu_2,\dots$ 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 $\nu_i$. For $\lambda>0$ denote $Z^\lambda_i:= \sum_x \lambda^x\nu_i(x)$ and $\lambda^{\max}:= \sup\{\lambda>0: Z^\lambda_i<\infty \text{ for all }i\}$, and assume $\lambda^{\max}>1$. For $\lambda<\lambda^{\max}$, define the tilted probability $\nu_i^{\lambda}(x):= \lambda^x\nu_i(x)/Z^{\lambda}_i$, and let $X^\lambda$ be a sequence of independent variables $X^\lambda_i$ with law $\nu^{\lambda}_i$, and denote $S^\lambda_n:= X^{\lambda}_1+\dots+X^{\lambda}_n$, with $S_n=S^1_n$. Choose $\lambda^*\in(1,\lambda^{\max})$ and denote $R^*_n:= E (S^{\lambda^*}_n)$. The Gibbs Conditioning Principle (GCP) holds if $P(X\in\cdot|S_n>R^*_n)$ converges weakly to the law of $X^{\lambda^*}$, as $n\to\infty$. We prove the GCP for log-concave $\nu_i$'s, meaning $\nu_i(x+1)\,\nu_i(x-1) \le ( \nu_i(x))^2$, 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 $\nu_i$'s, the canonical measures are stochastically ordered with respect to $k$. This, in turn, leads to the ordering of the conditioned tilted measures $P(X^\lambda\in\cdot|S^\lambda_n>R^*_n)$ in terms of $\lambda$. This ordering is a fundamental component of our proof.