A probabilistic approach to the Erd\"os-Kac theorem for additive functions

TL;DR

This paper introduces a probabilistic method to analyze the convergence rates of additive functions of random integers to Gaussian and Poisson distributions, broadening the scope beyond previous Fourier-based approaches.

Contribution

It provides new probabilistic bounds on distribution distances for additive functions, extending the Erd"os-Kac theorem to more general functions without Fourier analysis.

Findings

Bounds on Kolmogorov and Wasserstein distances to Gaussian distribution.

Bounds on Kolmogorov and total variation distances to Poisson distribution.

Generalization of existing results to broader classes of additive functions.

Abstract

We present a new perspective of assessing the rates of convergence to the Gaussian and Poisson distributions in the Erd\"os-Kac theorem for additive arithmetic functions $\psi$ of a random integer $J_n$ uniformly distributed over $\{1,...,n\}$. Our approach is probabilistic, working directly on spaces of random variables without any use of Fourier analytic methods, and our $\psi$ is more general than those considered in the literature. Our main results are (i) bounds on the Kolmogorov distance and Wasserstein distance between the distribution of the normalized $\psi(J_n)$ and the standard Gaussian distribution, and (ii) bounds on the Kolmogorov distance and total variation distance between the distribution of $\psi(J_n)$ and a Poisson distribution under mild additional assumptions on $\psi$. Our results generalize the existing ones in the literature.