Local limit theorem for complex valued sequences

TL;DR

This paper extends the local limit theorem to complex-valued sequences, providing detailed asymptotic behavior and convergence rates for iterated convolutions on the integer lattice.

Contribution

It generalizes the local limit theorem to complex sequences and establishes explicit convergence rates and asymptotic bounds.

Findings

Proves a sharp rate of convergence to an explicit attractor.

Derives a generalized Gaussian bound for asymptotic expansion.

Provides explicit formulas for asymptotic behavior of complex sequences.

Abstract

In this article, we study the pointwise asymptotic behavior of iterated convolutions on the one dimensional lattice Z. We generalize the so-called local limit theorem in probability theory to complex valued sequences. A sharp rate of convergence towards an explicitly computable attractor is proved together with a generalized Gaussian bound for the asymptotic expansion up to any order of the iterated convolution.