Mod-discrete expansions

TL;DR

This paper develops a method for approximating distributions of integer-valued random variables using mod-discrete expansions, especially when traditional convergence in law does not occur, with applications across probability, combinatorics, and number theory.

Contribution

It introduces a framework for approximating distributions via mod-discrete expansions based on characteristic function ratios, extending classical approximation techniques.

Findings

Applicable to classical probability, combinatorics, and number theory examples.

Provides conditions under which the ratio of characteristic functions converges.

Enables approximation of distributions without requiring convergence in law.

Abstract

In this paper, we consider approximating expansions for the distribution of integer valued random variables, in circumstances in which convergence in law cannot be expected. The setting is one in which the simplest approximation to the $n$'th random variable $X_n$ is by a particular member $R_n$ of a given family of distributions, whose variance increases with $n$. The basic assumption is that the ratio of the characteristic function of $X_n$ and that of R_n$ converges to a limit in a prescribed fashion. Our results cover a number of classical examples in probability theory, combinatorics and number theory.