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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*
University of Düsseldorf


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’.

Research Papers
Copyright © Applied Probability Trust 1978 

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[1] Bahadur, R. R. and Ranga Rao, R. (1960) On deviations of the sample mean. Ann. Math. Statist. 31, 10151027.CrossRefGoogle Scholar
[2] Book, S. A. (1976) Large deviation probabilities and the Erdôs–Rényi law of large numbers. Canad. J. Statist. 4, 2.Google Scholar
[3] Cox, D. R. (1962) Renewal Theory. Methuen, London.Google Scholar
[4] Cramér, H. (1938) Sur un nouveaux théorème-limite de la théorie des probabilités. Actualités Sci. Indust. No. 736, Hermann, Paris, 523.Google Scholar
[5] Erdös, P. (1949) On a theorem of Hsu and Robbins. Ann. Math. Statist. 20, 286291.Google Scholar
[6] Erdös, P. and Renyi, A. (1970) On a new law of large numbers. J. Analyse Math. 23, 103111.Google Scholar
[7] Lorden, G. (1970) On excess over the boundary. Ann. Math. Statist. 41, 520527.Google Scholar
[8] Parzen, A. (1962) Stochastic Processes. Holden-Day, San Francisco.Google Scholar
[9] Plachky, D. and Steinebach, J. (1975) A theorem about probabilities of large deviations with an application to queuing theory. Periodica Math. Hung. 6, 343345.Google Scholar
[10] Roberts, A. W. and Varberg, D. E. (1973) Convex Functions. Academic Press, New York.Google Scholar
[11] Takács, L. (1954) On secondary processes generated by a Poisson process and their applications in physics. Acta Math. Acad. Sci. Hung. 5, 203236.Google Scholar
[12] Takács, L. (1956) On secondary processes generated by recurrent processes. Acta Math. Acad. Sci. Hung. 7, 1729.Google Scholar
[13] Tsurui, A. and Osaki, S. (1976) On a first-passage problem for a cumulative process with exponential decay. Stoch. Proc. Appl. 4, 7988.CrossRefGoogle Scholar
[14] Wald, A. (1947) Sequential Analysis. Wiley, New York.Google Scholar