Total variation error bounds for geometric approximation

TL;DR

This paper introduces a novel Stein's method formulation to derive explicit total variation bounds for geometric approximation across various complex discrete distributions.

Contribution

It presents a new coupling-based framework for total variation bounds, applicable to diverse geometric-related distributions and models.

Findings

Derived explicit bounds for geometric approximation errors.

Applied the method to Galton-Watson processes and random graph models.

Demonstrated the approach's effectiveness in non-trivial examples.

Abstract

We develop a new formulation of Stein's method to obtain computable upper bounds on the total variation distance between the geometric distribution and a distribution of interest. Our framework reduces the problem to the construction of a coupling between the original distribution and the "discrete equilibrium" distribution from renewal theory. We illustrate the approach in four non-trivial examples: the geometric sum of independent, non-negative, integer-valued random variables having common mean, the generation size of the critical Galton-Watson process conditioned on non-extinction, the in-degree of a randomly chosen node in the uniform attachment random graph model and the total degree of both a fixed and randomly chosen node in the preferential attachment random graph model.