A unified approach to local limit theorems in Gaussian spaces and the law of small numbers

TL;DR

This paper introduces a unified, abstract framework for local limit theorems in Gaussian spaces by reformulating classical results using Wick products and second quantization, enabling broader applicability including infinite dimensions.

Contribution

It reformulates local limit theorems in Gaussian spaces using Wick products and second quantization, providing a general approach that extends to infinite-dimensional spaces.

Findings

Established L^1 convergence of densities to the limit

Extended the framework to infinite-dimensional Gaussian spaces

Unified classical limit theorems within a new abstract setting

Abstract

Through a reformulation of the local limit theorem and law of small numbers, which is obtained by working in the spaces naturally associated to the limiting distributions, we discover a general and abstract framework for the investigation of that type of limit theorems. From this new perspective, the convolution and scaling operators utilized in the classical results mentioned before will be identified with the Wick product and second quantization operators, respectively. And here is the advantage of our approach: definitions and most of the properties of Wick products and second quantization operators do not depend (mutatis mutandis) on the underlying probability measure. Then, with the help of Holder-Young-type inequalities for Gaussian and Poisson Wick products proved in previous papers, we show the L^1 convergence of the densities towards the desired limit. We remark that our approach extends without additional assumptions to infinite dimensional Gaussian spaces.