Interval Type Local Limit Theorems for Lattice Type Random Variables and Distributions

TL;DR

This paper introduces a new interpretation of local limit theorems for lattice distributions, showing uniform approximation on intervals and identifying decay speed limits, with applications to sums of i.i.d. and correlated vectors.

Contribution

It provides a novel framework for interval type local limit theorems, including continuous distributions, and characterizes decay speed restrictions for lattice distributions.

Findings

Distributions are well approximated by the limit uniformly on intervals.

Identifies the maximum decay speed of interval lengths for approximation.

Continuous distributions satisfy the local law without decay restrictions.

Abstract

In this paper, we propose a new interpretation of local limit theorems for univariate and multivariate distributions on lattices. We show that - given a local limit theorem in the standard sense - the distributions are approximated well by the limit distribution, uniformly on intervals of possibly decaying length. We identify the maximally allowable decay speed of the interval lengths. Further, we show that for continuous distributions, the interval type local law holds without any decay speed restrictions on the interval lengths. We show that various examples fit within this framework, such as standardized sums of i.i.d. random vectors or correlated random vectors induced by multidimensional spin models from statistical mechanics.