Intertwinings and Stein's magic factors for birth-death processes

TL;DR

This paper develops second order intertwinings for birth-death processes to derive new bounds on Stein factors, enhancing distribution approximation techniques for discrete distributions.

Contribution

It introduces second order intertwining relations for birth-death semigroups, extending previous first order results and enabling improved Stein factor bounds.

Findings

New quantitative bounds on Stein factors for discrete distributions

Enhanced approximation results for Poisson and geometric mixtures

Extension of intertwining techniques to second derivatives

Abstract

This article investigates second order intertwinings between semigroups of birth-death processes and discrete gradients on the space of natural integers N. It goes one step beyond a recent work of Chafa{\"i} and Joulin which establishes and applies to the analysis of birth-death semigroups a first order intertwining. Similarly to the first order relation, the second order intertwining involves birth-death and Feynman-Kac semigroups and weighted gradients on N, and can be seen as a second derivative relation. As our main application, we provide new quantitative bounds on the Stein factors of discrete distributions. To illustrate the relevance of this approach, we also derive approximation results for the mixture of Poisson and geometric laws.