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A strong law of Erdös-Rényi type for cumulative processes in renewal theory

Published online by Cambridge University Press:  14 July 2016

Josef Steinebach*
Affiliation:
University of Düsseldorf

Abstract

Let {Nt}t >0 be a renewal counting process (cf. Parzen (1962), p. 160) with underlying failure times let be a sequence of non-negative random variables and {Zt}t >0 an associated cumulative process, i.e. if Nt = 1, 2, …, and Zt = 0, if Nt = 0. By convention set Z0 = 0. Consider the maximum increment of the process {Zt}t >0 in [0, T] over a time K, 0 < K < T, divided by K. Under appropriate conditions it is shown that for a wide range of numbers a there exist constants C(a), uniquely determined by a and the distributions of the Xi's and Yj's, such that D(T, C log T) converges to a with probability 1. This result provides a renewal theoretic variant of Erdös and Rényi's (1970) ‘new law of large numbers’.

Type
Research Papers
Copyright
Copyright © Applied Probability Trust 1978 

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