A SHARP CORRELATION INEQUALITY WITH APPLICATION TO
ALMOST SURE LOCAL LIMIT THEOREM
Abstract: 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 with ). The approach
used to treat the case breaks down when . MacDonald’s concept of
the Bernoulli part of a random variable is used in a crucial way to remedy this.
2000 AMS Mathematics Subject Classification: Primary: 60F15, 60G50; Secondary:
60F05.
Keywords and phrases: Correlation inequality, i.i.d. random variables, lattice
distributed, Bernoulli part, square integrable, local limit theorem, almost sure version.