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A bound on the rate of convergence in the central limit theorem for renewal processes under second moment conditions

Part of: Limit theorems Stochastic processes

Published online by Cambridge University Press:  04 May 2020

G. Reinert  and
C. Yang
G. Reinert*
Affiliation:
University of Oxford
C. Yang*
Affiliation:
University of Waterloo
*
*Postal address: Department of Statistics, University of Oxford, 24-29 St Giles’, Oxford, OX1 3LB, UK. Email address: [email protected]
**Postal address: Department of Statistics and Actuarial Science, University of Waterloo, 200 University Avenue West, Waterloo, Ontario, CanadaN2L 3G1. Email address: [email protected]
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Abstract

A famous result in renewal theory is the central limit theorem for renewal processes. Since, in applications, usually only observations from a finite time interval are available, a bound on the Kolmogorov distance to the normal distribution is desirable. 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–Esseen theorem under second moment conditions.

Keywords

rate of convergencecentral limit theoremStein’s method

MSC classification

Primary: 60F05: Central limit and other weak theorems
Secondary: 60G50: Sums of independent random variables; random walks
Type
Research Papers
Information
Journal of Applied Probability , Volume 57 , Issue 1 , March 2020 , pp. 343 - 360
Copyright
© Applied Probability Trust 2020

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