Hostname: page-component-8448b6f56d-m8qmq Total loading time: 0 Render date: 2024-04-23T20:36:39.400Z Has data issue: false hasContentIssue false
  • English
  • Français

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: reinert@stats.ox.ac.uk
**Postal address: Department of Statistics and Actuarial Science, University of Waterloo, 200 University Avenue West, Waterloo, Ontario, CanadaN2L 3G1. Email address: c298yang@uwaterloo.ca
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Ahmad, I. A. (1981). The exact order of normal approximation in bivariate renewal theory. Adv. Appl. Prob. 13, 113–128.CrossRefGoogle Scholar
Arratia, R., Martin, D., Reinert, G. and Waterman, M. S. (1996). Poisson process approximation for sequence repeats, and sequencing by hybridization. J. Comput. Biol. 3, 425–463.CrossRefGoogle ScholarPubMed
Billingsley, P. (1968). Convergence of Probability Measures. John Wiley, New York.Google Scholar
Chen, L. H. Y, Goldstein, L., andShao, Q.-M. (2011). Normal Approximation by Stein’s Method. Springer, Heidelberg.CrossRefGoogle Scholar
Chen, L. H. Y., & Shao, Q. M. (2001). A non-uniform Berry-–Esseen bound via Stein’s method. Prob. Theory Relat. Fields 120, 236–254.CrossRefGoogle Scholar
Chen, L. H., & Shao, Q. M. (2005). Stein’s method for normal approximation. In: An Introduction to Stein’s Method, eds A. D. Barbour and L. H. Y. Chen, World Scientific, Singapore, pp. 1–59.Google Scholar
Chernoff, H. (2004). Conservative bounds on extreme P-values for testing the equality of two probabilities based on very large sample sizes. In A Festschrift for Herman Rubin, ed A. DasGupta, Institute of Mathematical Statistics, Beachwood, OH, pp. 250–254.CrossRefGoogle Scholar
Englund, G. (1980). Remainder term estimate for the asymptotic normality of the number of renewals. J. Appl. Prob. 17, 1108–1113.CrossRefGoogle Scholar
Niculescu, S. P., & Omey, E. (1994). Asymptotic behaviour of the distribution function of multivariate nonhomogeneous renewal processes. Studia Sci. Math. Hung. 29, 179–190.Google Scholar
Omey, E., & Vesilo, R. (2016). Local limit theorems for shock models. Brazilian J. Prob. Statist. 30, 221–247.CrossRefGoogle Scholar
Roginsky, A. L. (1992). A central limit theorem by remainder term for renewal processes. Adv. Appl. Prob. 24, 267–287.CrossRefGoogle Scholar
Stein, C. (1972). A bound for the error in the normal approximation to the distribution of a sum of dependent random variables. In Proc. 6th Berkeley Symp. Math. Statist. Prob., Vol. 2, University of California Press, Berkeley, CA.Google Scholar