Mod-$\phi$ convergence: Approximation of discrete measures and harmonic analysis on the torus

TL;DR

This paper develops a Fourier-analytic framework for approximating lattice-distributed random variables using mod-$$ convergence, enabling better discrete distribution approximations and applications in combinatorics and number theory.

Contribution

It introduces a novel approach using signed measures and harmonic analysis to improve approximation schemes for lattice distributions, extending to higher dimensions.

Findings

Enhanced approximation accuracy over standard Poisson methods

Applicable to diverse combinatorial and number-theoretic problems

Explicitly accounts for correlations in multivariate settings

Abstract

In this paper, we relate the framework of mod-$\phi$ convergence to the construction of approximation schemes for lattice-distributed random variables. The point of view taken here is that of Fourier analysis in the Wiener algebra, allowing the computation of asymptotic equivalents in the local, Kolmogorov and total variation distances. By using signed measures instead of probability measures, we are able to construct better approximations of discrete lattice distributions than the standard Poisson approximation. This theory applies to various examples arising from combinatorics and number theory: number of cycles in (possibly coloured) permutations, number of prime divisors (possibly within different residue classes) of a random integer, number of irreducible factors of a random polynomial, etc. One advantage of the approach developed in this paper is that it allows us to deal with approximations in higher dimensions as well. In this setting, we can explicitly see the influence of the correlations between the components of the random vectors in our asymptotic formulas.