Gibbs Conditioning Principle for Log-concave Independent Random Variables

Let ν1, ν2, . . . be a sequence of probabilities on the nonnegative integers, and X = (X1, X2, . . .) be a sequence of independent random variables Xi with law νi. For λ > 0 denote Zλ i := ∑ x λxνi(x) and λmax := sup{λ > 0 : Zλ i < ∞ for all i}, and assume λmax > 1. For λ < λmax, define the tilted probability νλ i (x) := λxνi(x)/Zλ i , and let Xλ be a sequence of independent variables Xλ i with law νλ i , and denote Sλ n := Xλ 1 + · · · + Xλ n , with Sn = S1 n. Choose λ∗ ∈ (1, λmax) and denote R∗ n := E(Sλ∗n ). The Gibbs Conditioning Principle (GCP) holds if P (X ∈ · |Sn > R∗n) converges weakly to the law of Xλ∗, as n → ∞. We prove the GCP for log-concave νi’s, meaning νi(x + 1) νi(x − 1) ≤ (νi(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 νi’s, the canonical measures are stochastically ordered with respect to k. This, in turn, leads to the ordering of the conditioned tilted measures P (Xλ ∈ · |Sλn > R∗n) in terms of λ. This ordering is a fundamental component of our proof.