  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.
