A bound on the rate of convergence in the central limit theorem for renewal processes under second moment conditions. (March 2020)

• Record Type: Journal Article
• Authors: Reinert, G.; Yang, C.
• Is Part Of: Journal of applied probability. Volume 57:Number 1(2020)
• Page Start: 343
• Page End: 360
• Publication Date: 2020-03
• Subjects: rate of convergence, -- central limit theorem, -- Stein's method; 60F05, -- 60G50; 519.2
• DOI: 10.1017/jpr.2019.101
• Languages: English
• ISSNs: 0021-9002

Abstract

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.