New rates for exponential approximation and the theorems of R\'{e}nyi and Yaglom

TL;DR

This paper develops new theoretical tools to improve exponential approximation rates in probability, applying them to classical theorems like Rényi's and Yaglom's, using Stein's method and coupling techniques.

Contribution

It introduces two abstract theorems that simplify complex exponential approximation problems to coupling constructions, providing new convergence rates for key probabilistic theorems.

Findings

New convergence rates for Rényi's theorem on random sums

Improved rates for hitting times in Markov chains

A novel rate for Yaglom's theorem on Galton--Watson processes

Abstract

We introduce two abstract theorems that reduce a variety of complex exponential distributional approximation problems to the construction of couplings. These are applied to obtain new rates of convergence with respect to the Wasserstein and Kolmogorov metrics for the theorem of R\'{e}nyi on random sums and generalizations of it, hitting times for Markov chains, and to obtain a new rate for the classical theorem of Yaglom on the exponential asymptotic behavior of a critical Galton--Watson process conditioned on nonextinction. The primary tools are an adaptation of Stein's method, Stein couplings, as well as the equilibrium distributional transformation from renewal theory.