A sharp correlation inequality with an application to almost sure local limit theorem

Abstract

We prove a new sharp correlation inequality for sums of i.i.d. square integrable lattice distributed random variables. We also apply it to establish an almost sure local limit theorem for iid square integrable random variables taking values in an arbitrary lattice. This extends a recent similar result jointly obtained with Antonini-Giuliano, under a slightly stronger absolute moment assumption (of order $2+u$ with $u>0$). The approach used to treat the case $u>0$ breaks down when $u=0$. MacDonald's concept of the Bernoulli part of a random variable is used in a crucial way to remedy this.

Cited by in corpus (3)

• A general correlation inequality and the Almost Sure Local Limit Theorem for random sequences in the domain of attraction of a stable law
• A Borel-Cantelli lemma and its applications
• Almost Sure Local Limit Theorem for the Dickman distribution