A nonconventional local limit theorem

TL;DR

This paper extends local limit theorem results to nonconventional sums involving functions of multiple scaled indices of a stochastic process, broadening the scope of classical limit theorems.

Contribution

It introduces a new approach to local limit theorems for sums of functions of nonconventional index sequences, which was not previously addressed.

Findings

Established local limit theorems for nonconventional sums

Extended classical results to functions of multiple scaled indices

Provided conditions under which the theorems hold

Abstract

We extend local limit theorem type results to nonconventional sums of the form $S_N=\sum_{n=1}^NF(\xi_n,\xi_{2n},...,\xi_{\ell n})$.