Gibbs Conditioning Principle for Log-concave Independent Random Variables

E. Cator, Pablo A. Ferrari

2026, v.32, Issue 1, 63-77

ABSTRACT

Let ν_1, ν_2, . . . be a sequence of probabilities on the nonnegative integers, and X = (X_1, X_2, . . .) be a sequence of independent random variables X_i with law ν_i. For λ > 0 denote Z^λ_i := ∑_x λ^x ν_i(x) and λ^max := sup{λ > 0 : Z^λ_i < ∞ for all i}, and assume λ^max > 1. For λ < λ^max, define the tilted probability ν^λ_i (x) := λ^x ν_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 + · · · + X^λ_n , with S_n = S^1_n. Choose λ* ∈ (1, λ^max) and denote R*_n := E(S^(λ*)_n). The Gibbs Conditioning Principle (GCP) holds if P (X ∈ · |S_n > R*_n) converges weakly to the law of X^(λ*), as n → ∞. We prove the GCP for log-concave ν_i’s, meaning ν_i(x + 1) ν_i(x − 1) ≤ (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 ∈ · |S^λ_n > R*_n) in terms of λ. This ordering is a fundamental component of our proof.

doi:10.61102/1024-2953-mprf.2026.32.1.003

Keywords: empirical measures, Gibbs conditioning principle, large deviations, log-concave random variables, Efron theorem

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