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 $Z_i^{\lambda }:={\sum }_x{\lambda }^x{\nu }_i(x)$ and ${\lambda }^{\max }:=\sup \{\lambda >0:Z_i^{\lambda }<\mathrm{\infty }\text{ for all }i\}$, and assume ${\lambda }^{\max }>1$. For $\lambda <{\lambda }^{\max }$, define the tilted probability ${\nu }_i^{\lambda }(x):={\lambda }^x{\nu }_i(x)\mathrm{/}{Z}_i^{\lambda }$, and let $X^{\lambda }$ be a sequence of independent variables $X_i^{\lambda }$ with law ${\nu }_i^{\lambda }$, and denote $S_n^{\lambda }:=X_1^{\lambda }+\cdots +X_n^{\lambda }$, with $S_n=S_n^1$. Choose ${\lambda }^{\ast }\in (1,{\lambda }^{\max })$ and denote $R_n^{\ast }:=E(S_n^{{\lambda }^{\ast }})$. The Gibbs Conditioning Principle (GCP) holds if $P(X\in \cdot \mathrm{\mid }{S}_n>R_n^{\ast })$ converges weakly to the law of $X^{{\lambda }^{\ast }}$, as $n\to \mathrm{\infty }$. 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 $P(X^{\lambda }\in \cdot \mathrm{\mid }{S}_n^{\lambda }>R_n^{\ast })$ in terms of $\lambda$. This ordering is a fundamental component of our proof.