16:00-17:00 Thursday, 8 May 2025 Auditorium, Rectorate
Abstract
Denote $\mu$ the distribution of a family of independent nonnegative integer random variables $X_i$, with marginals $\nu_i$. Assume that the means $m_i$ converge to some real number $m$, and take $R>m$. Let $\overline X_n$ be the empirical mean of the first $n$ variables. Define the probability measures $\nu_i^{\lambda}(x)= \lambda^x\nu_i(x)/Z_\lambda$, and denote its mean by $m^\lambda_i$. Denote $\lambda(R)$ the $\lambda$ satisfying $m^{\lambda}_i\to_i R$. The Gibbs conditioning principle (GCP) says that as $n\to\infty$, $\mu(\cdot|\overline X_n>R)$ converges weakly to the product measure with marginals $\nu^{\lambda(R)}_i$. Assuming that $\lambda(R)$ exists and is finite, we prove the GCP under the log-concave condition $\nu_i(x+1),\nu_i(x-1) \le ( \nu_i(x))^2$. Joint with Eric Cator.