An application of the Bernoulli part to local limit theorems for moving averages on stationary sequences

We consider partial sums S n of a general class of stationary sequences of integer‐valued random variables, and we provide sufficient conditions for S n to satisfy a local limit theorem. To prove this result, we introduce a concept called the Bernoulli part. The amount of Bernoulli part in S n determines the extent to which the density of S n is relatively flat. If in addition S n satisfies a global central limit theorem, the local limit theorem follows.