Local Limit Theorems for Complex Functions on $\mathbb{Z}^d$

TL;DR

This paper extends local limit theorems for complex functions on $ abla^d$ by including cases where the Fourier transform peaks at imaginary points, resulting in oscillatory attractors and broadening the understanding of convolution powers.

Contribution

It generalizes previous local limit theorems to include complex-valued functions with Fourier maxima at imaginary points, introducing oscillatory attractors and advanced analytical techniques.

Findings

Established local limit theorems with oscillatory attractors for complex functions.

Extended sup-norm estimates for these functions.

Used generalized polar coordinates and Van der Corput lemma for proofs.

Abstract

The local (central) limit theorem precisely describes the behavior of iterated convolution powers of a probability distribution on the $d$-dimensional integer lattice, $\mathbb{Z}^d$. Under certain mild assumptions on the distribution, the theorem says that the convolution powers are well-approximated by a single scaled Gaussian density which we call an attractor. When such distributions are allowed to take on complex values, their convolution powers exhibit new and disparate behaviors not seen in the probabilistic setting. Following works of I. J. Schoenberg, T. N. E. Greville, P. Diaconis, and L. Saloff-Coste, the author and L. Saloff-Coste provided a complete description of local limit theorems for the class of finitely supported complex-valued functions on $\mathbb{Z}$. For convolution powers of complex-valued functions on $\mathbb{Z}^d$, much less is known. In a previous work by the author and L. Saloff-Coste, local limit theorems were established for complex-valued functions whose Fourier transform is maximized in absolute value at so-called points of positive homogeneous type and, in that case, the resultant attractors are generalized heat kernels corresponding to a class of higher order partial differential operators. By considering the possibility that the Fourier transform can be maximized in absolute value at points of imaginary homogeneous type, this article extends previous work of the author and L. Saloff-Coste to broaden the class of complex-valued functions for which it is possible to obtain local limit theorems. These local limit theorems contain attractors given by certain oscillatory integrals and their convergence is established using a generalized polar-coordinate integration formula, due to H. Bui and the author, and the Van der Corput lemma. The article also extends recent results on sup-norm type estimates of H. Bui and the author.