A SHARP CORRELATION INEQUALITY WITH APPLICATION TO ALMOST SURE LOCAL LIMIT THEOREM

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 version of the local limit theorem for i.i.d. square integrable random variables taking values in an arbitrary lattice. This extends a recent similar result jointly obtained with Giuliano-Antonini 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. Mac-Donald's concept of the Bernoulli part of a random variable is used in a crucial way to remedy this.