Mod-Gaussian convergence from a factorisation of the probability generating function

Abstract

Let $(X_{n})_{n\in \mathbb{N}}$ be a sequence of bounded random variables with nonnegative integer values, and such that $\sigma _{n}^{2}:=\mathop{ \mathrm{Var}}\nolimits (X_{n})$ tends to infinity. Denote $P_{n}(t) = \mathbb{E}[t^{X_{n}}]$ the probability generating function of X _n . Each P _n (t) is a polynomial in t. Then, it is known that a sufficient condition for X _n to be asymptotically Gaussian is that P _n (t) has negative real roots (see references below). In this chapter, we prove that if the third cumulant L _n ³: = κ ⁽³⁾(X _n ) also tends to infinity with light additional hypotheses, then a suitable renormalised version of X _n converges in the mod-Gaussian sense. We then give an application for the number of blocks in a uniform set-partition of [n].