Abstract
Let X-m+1, X-m+2,..., X0, X1, X2,..., Xn be a time-homogeneous {0, 1}-valued m-th order Markov chain. The probability distributions of numbers of runs of "1" of length k (k ≥ m) and of "1" of length k (k < m) in the sequence of a {0, 1}-valued m-th order Markov chain are studied. There are some ways of counting numbers of runs with length k. This paper studies the distributions based on four ways of counting numbers of runs, i.e., the number of non-overlapping runs of length k, the number of runs with length greater than or equal to k, the number of overlapping runs of length k and the number of runs of length exactly k.
Similar content being viewed by others
REFERENCES
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. Havakawa), 467–474, VSP International Science Publishers, Zeist.
Aki, S. and Hirano, K. (1996). Lifetime distribution and estimation problems of consecutive-k-out-of-n:F systems, Ann. Inst. Statist. Math., 48, 185–199.
Chao, M. T., Fu, J. C. and Koutras, M. V. (1995). A survey of the reliability studies of consecutive-k-out-of-n:F systems and its related systems, IEEE Transactions on Reliability, 44, 120–127.
Fu, J. C. (1996). Distribution theory of runs and patterns associated with a sequence of multistate trials, Statist. Sinica, 6, 957–974.
Fu, J. C. and Koutras, M. V. (1994). Distribution theory of runs: A Markov chain approach, J. Amer. Statist. Assoc., 89, 1050–1058.
Goldstem, L. (1990). Poisson approximation and DNA sequence matching, Comm. Statist. Theory Methods, 19, 4167–4179.
Hearn, A. C. (1993). REDUCE User's Manual, Ver. 3.5, Rand Publication CP78 (Rev. 10/93), Konrad-Zuse-Zentrum, Berlin.
Hirano, K. (1986). Some properties of the distributions of order k, Fibonacci Numbers and Their Applications (eds. A. N. Philippou, G. E. Bergum and A. F. Horadam), 43–53, Reidel, Dordrecht.
Hirano, K. (1994). Consecutive-k-out-of-n:F systems, Proc. Inst. Statist. Math., 42, 45–61 (in Japanese).
Hirano, K. and Aki, S. (1993). On number of occurrences of success runs of specified length in a two-state Markov chain, Statist. Sinica. 3, 313–320.
Hirano, K., Aki, S. and Uchida, M. (1997). Distributions of numbers of success-runs until the first consecutive k successes in higher order Markov dependent trials, Advances in Combinatorial Methods and Applications to Probability and Statistics (ed. N. Balakrishnan), 401–410, Birkhauser, Boston.
Koutras, M. V. and Alexandrou, V. A. (1995). Runs, scans and urn model distributions: A unified Markov chain approach, Ann. Inst. Statist. Math., 47, 743–766.
Ling, K. D. (1988). On binomial distributions of order k, Statist. Probab. Lett., 6, 247–250.
Ling, K. D. (1989). A new class of negative binomial distributions of order k, Statist. Probab. Lett., 7, 371–376.
Mood, A. M. (1940). The distribution theory of runs, Ann. Math. Statist., 11, 367–392.
Philippou, A. N. and Makri, F. S. (1986). Successes, runs and longest runs, Statist. Probab. Lett., 4, 101–105.
Stanley, R. P. (1986). Enumerative Combinatorics, Vol. I, Wadsworth & Brooks/Cole, California.
Uchida, M. (1994). Order statistics with discrete distributions and probability that ties occur somewhere, J. Japan Soc. Comput. Staist., 7, 47–56.
Uchida, M. (1996a). On number of occurrences of success-runs of exact length k, Research Memo., No. 597, The Institute of Statistical Mathematics, Tokyo.
Uchida, M. (1996b). Joint distributions of numbers of success-runs until the first consecutive k successes in a higher-order two-state Markov chain, Research Memo., No. 619, The Institute of Statistical Mathematics, Tokyo.
Uchida, M. (1996c). On number of occurrences of success runs specified length in a higher-order two-state Markov chain, Research Memo., No. 625, The Institute of Statistical Mathematics, Tokyo.
Uchida, M. and Aki, S. (1995). Sooner and later waiting time problems in a two-state Markov chain, Ann. Inst. Statist. Math., 47, 415–433.
Author information
Authors and Affiliations
About this article
Cite this article
Uchida, M. On Number of Occurrences of Success Runs of Specified Length in a Higher-Order Two-State Markov Chain. Annals of the Institute of Statistical Mathematics 50, 587–601 (1998). https://doi.org/10.1023/A:1003537831046
Issue Date:
DOI: https://doi.org/10.1023/A:1003537831046