Abstract
Let X 1, X 2,... be a sequence of nonnegative integer valued random variables.For each nonnegative integer i, we are given a positive integer k i . For every i = 0, 1, 2,..., E i denotes the event that a run of i of length k i occurs in the sequence X 1, X 2,.... For the sequence X 1, X 2,..., the generalized pgf's of the distributions of the waiting times until the r-th occurrence among the events % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiiYdd9qrFfea0dXdf9vqai-hEir8Ve% ea0de9qq-hbrpepeea0db9q8as0-LqLs-Jirpepeea0-as0Fb9pgea% 0lrP0xe9Fve9Fve9qapdbaqaaeGacaGaaiaabeqaamaabaabcaGcba% WaaiWabeaacaWGfbWaaSbaaSqaaiaadMgaaeqaaaGccaGL7bGaayzF% aaWaa0baaSqaaiaadMgacqGH9aqpcaaIWaaabaGaeyOhIukaaaaa!43D8!\[\left\{ {E_i } \right\}_{i = 0}^\infty\]are obtained. Though our situations are general, the results are very simple. For the special cases that X's are i.i.d. and {0, 1}-valued, the corresponding results are consistent with previously published results.
Similar content being viewed by others
References
Aki, S. (1985). Discrete distribution of order k on a binary sequence, Ann. Inst. Statist. Math., 37, 205–224.
Aki, S. and Hirano, K. (1989). Estimation of parameters in the discrete distributions of order k, Ann. Inst. Statist. Math., 41, 47–61.
Aki, S., Kuboki, H. and Hirano, K. (1984). On discrete distributions of order k, Ann. Inst. Statist. Math., 36, 431–440.
Ebneshahrashoob, M. and Sobel, M. (1990). Sooner and later waiting time problems for Bernoulli trials: frequency and run quotas, Statist. Probab. Lett., 9, 5–11.
Feller, W. (1957). An Introduction to Probability Theory and Its Applications, Vol. 1, 2nd ed., Wiley.
Ling, K. D. (1990). On geometric distributions of order (k 1,..., k m ), Statist. Probab. Lett., 9, 163–171.
Philippou, A. N. (1986). Distributions and Fibonacci polynomials of order k, longest runs, and reliability of consecutive-k-out-of-n:F systems, Fibonacci Numbers and Their Applications (eds. A. N. Philippou, G. E. Bergum and A. F. Horadam), 203–227, Reidel, Dordrecht.
Philippou, A. N., Georghiou, C. and Philippou, G. N. (1983). A generalized geometric distribution and some of its properties, Statist. Probab. Lett., 1, 171–175.
Author information
Authors and Affiliations
Additional information
This research was partially supported by the ISM Cooperative Research Program (90-ISM-CRP-11) of the Institute of Statistical Mathematics.
About this article
Cite this article
Aki, S. Waiting time problems for a sequence of discrete random variables. Ann Inst Stat Math 44, 363–378 (1992). https://doi.org/10.1007/BF00058646
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF00058646