A Bound on the Rate of Convergence in the Central Limit Theorem for Renewal Processes under Second Moment Conditions

TL;DR

This paper derives explicit bounds on the convergence rate to normality in the renewal process CLT under second moment conditions, using Stein's method, with practical constants and intermediate Berry-Essen bounds.

Contribution

It provides the first explicit non-uniform bound for the renewal CLT under minimal second moment assumptions, tracking explicit constants.

Findings

Explicit non-uniform bound for renewal CLT

Explicit bounds in Berry-Essen theorem under second moment conditions

Constants explicitly tracked in the bounds

Abstract

A famous result in renewal theory is the Central Limit Theorem for renewal processes. As in applications usually only observations from a finite time interval are available, a bound on the Kolmogorov distance to the normal distribution is desirable. Here we provide an explicit non-uniform bound for the Renewal Central Limit Theorem based on Stein's method and track the explicit values of the constants. For this bound the inter-arrival time distribution is required to have only a second moment. As an intermediate result of independent interest we obtain explicit bounds in a non-central Berry-Ess\'{e}n theorem under second moment conditions.