Abstract
LetX 1,X 2,... be a time-homogeneous {0, 1}-valued Markov chain. LetF 0 be the event thatl runs of “0” of lengthr occur and letF 1 be the event thatm runs of “1” of lengthk occur in the sequenceX 1,X 2, ... We obtained the recurrence relations of the probability generating functions of the distributions of the waiting time for the sooner and later occurring events betweenF 0 andF 1 by the non-overlapping way of counting and overlapping way of counting. We also obtained the recurrence relations of the probability generating functions of the distributions of the sooner and later waiting time by the non-overlapping way of counting of “0”-runs of lengthr or more and “1”-runs of lengthk or more.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Aki, S. (1992). Waiting time problems for a sequence of discrete random variables,Ann. Inst. Statist. Math.,44, 363–378.
Aki, S. and Hirano, K. (1989). Estimation of parameters in the discrete distributions of orderk, Ann. Inst. Statist. Math.,41, 47–61.
Aki, S. and Hirano, K. (1993). Discrete distributions related to succession events in a two-state Markov chain,Statistical Sciences and Data Analysis; Proceedings of the Third Pacific Area Statistical Conference (eds. K. Matusita, M. L. Puri and T. Hayakawa), 467–474, VSP International Science Publishers, Zeist.
Balasubramanian, K., Viveros, R. and Balakrishnan, N. (1993). Sooner and later waiting time problems for Markovian Bernoulli trials,Statist. Probab. Lett.,18, 153–161.
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. (1968).An Introduction to Probability Theory and Its Applications, Vol. I, 3rd ed., Wiley, New York.
Fu, J. C. and Koutras, M. V. (1994a). Poisson approximations for 2-dimensional patterns,Ann. Inst. Statist. Math.,46, 179–192.
Fu, J. C. and Koutras, M. V. (1994b). Distribution theory of runs: a Markov chain approach,J. Amer. Statist. Assoc.,89, 1050–1058.
Godbole, A. P. (1993). Approximate reliabilities ofm-consecutive-k-out-of-n: failure systems,Statist. Sinica,3, 321–327.
Goldstein, L. (1990). Poisson approximation and DNA sequence matching,Comm. Statist. Theory Methods,19, 4167–4179.
Hearn, A. C. (1984).REDUCE User's Manual, Version 3.1, The Rand Corporation, Santa Monica.
Hirano, K., Aki, S., Kashiwagi, N. and Kuboki, H. (1991). On Ling's binomial and negative binomial distributions of orderk, Statist. Probab. Lett.,11, 503–509.
Koutras, M. V. and Papastavridis, S. G. (1983). On the number of runs and related statistics,Statist. Sinica,3, 277–294.
Ling, K. D. (1988). On binomial distributions of orderk, Statist. Probab. Lett.,6, 247–250.
Ling, K. D. (1989). A new class of negative binomial distributions of orderk, Statist. Probab. Lett.,7, 371–376.
Mohanty, S. G. (1994). Success runs of lengthk in Markov dependent trials,Ann. Inst. Statist. Math.,46, 777–796.
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.
Stanley, R. P. (1986).Enumerative Combinatorics, Vol. I, Wadsworth & Brooks/Cole, California.
Author information
Authors and Affiliations
About this article
Cite this article
Uchida, M., Aki, S. Sooner and later waiting time problems in a two-state Markov chain. Ann Inst Stat Math 47, 415–433 (1995). https://doi.org/10.1007/BF00773392
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF00773392