Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2024-12-23T17:48:59.617Z Has data issue: false hasContentIssue false
  • English
  • Français

On some statistics connected with runs in Markov chains

Published online by Cambridge University Press:  14 July 2016

Dragan Banjević
Dragan Banjević*
Affiliation:
University of Belgrade
*
∗Postal address: Institut za Matematiku, Pirodno-matematički fakultet, Studentski trg 16, 11000 Belgrade, Yugoslavia.
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider a class of statistics connected with the history of a Markov chain before the first occurrence of certain runs. Methods for deriving generating functions, distributions and moments of the statistics considered are given. In the case of i.i.d. random variables, explicit formulas are derived.

Keywords

SEQUENCE PATTERNSWAITING TIMESTOPPING TIMEOCCURRENCE OF RUNS
Type
Short Communications
Information
Journal of Applied Probability , Volume 25 , Issue 4 , December 1988 , pp. 815 - 821
Copyright
Copyright © Applied Probability Trust 1988 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Blom, G. (1982) On the mean number of random digits until a given sequence occurs. J. Appl. Prob. 19, 136–143.CrossRefGoogle Scholar
Blom, G. (1984) On the stochastic ordering of waiting times for patterns in sequences of random digits. J. Appl. Prob. 21, 730–737.CrossRefGoogle Scholar
Blom, G. and Thorburn, D. (1982) How many random digits are required until given sequences are obtained? J. Appl. Prob. 19, 518–531.CrossRefGoogle Scholar
Chen, R. and Zame, A. (1979) On fair coin-tossing games. J. Multivariate Anal. 9, 150–156.CrossRefGoogle Scholar
Feller, W. (1968) An Introduction to Probability Theory and its Applications , Vol. 1. Wiley, New York.Google Scholar
Gerber, H. U. and Li, S. R. (1981) The occurrence of sequence patterns in repeated experiments and hitting times in a Markov chain. Stoch. Proc. Appl. 11, 101–108.CrossRefGoogle Scholar
Glaz, J. (1983) Moving window detection for discrete data. IEEE Trans. Inform. Theory IT29, 457–462.CrossRefGoogle Scholar
Glick, I. I. (1981) A note on runs. Z. Angew. Math. Mech. 61, 119–120.CrossRefGoogle Scholar
Greenberg, I. (1970) The first occurrence of n successes in N trials. Technometrics 12, 627–634.CrossRefGoogle Scholar
Guibas, L. J. and Odlyzko, A. M. (1980) Long repetitive patterns in random sequences. Z. Wahrscheinlichkeitsch. 53, 241–262.CrossRefGoogle Scholar
Guibas, L. J. and Odlyzko, A. M. (1981) String overlaps, pattern matching, and nontransitive games. J. Combin. Theory A30, 183–208.CrossRefGoogle Scholar
Huntington, R. J. (1976) Mean recurrence times for k successes within m trials. J. Appl. Prob. 13, 604–607.CrossRefGoogle Scholar
Ivanov, V. A. and Novikov, A. E. (1977) On the distribution of the time up to the first occurrence of a given number of different l-tuple series. Teor. Veroyatnost. i Primenen. 22, 546–555.Google Scholar
Johnson, N. L. (1968) Repetitions. Amer. Math. Monthly 75, 382–383.CrossRefGoogle Scholar
Johnson, N. L. (1976) A return to repetitions, in Essays in Probability and Statistics , Shinko Fsusho, Tokyo, 635–644.Google Scholar
Li, S. R. (1980) A martingale approach to the study of occurrence of sequence patterns in repeated experiments. Ann. Prob. 8, 1171–1176.CrossRefGoogle Scholar
Móri, T. F. (1985) Large deviation results for waiting times in repeated experiments. Acta Math. Hung. 45, 213–221.CrossRefGoogle Scholar
Móri, T. F. and Szekely, G. J. (1984) Asymptotic independence of ‘pure head’ stopping times. Statist. Prob. Lett. 2, 5–8.CrossRefGoogle Scholar
Nielsen, P. T. (1973) On the expected duration of a search for a fixed pattern in random data. IEEE Trans. Inform. Theory IT19, 702–704.CrossRefGoogle Scholar
Ràde, L. (1976) Waiting for patterns in a sequence of random numbers. Z. Angew. Math. Mech. 56, 124–125.CrossRefGoogle Scholar
Reza Soltani, A. and Khodadadi, A. (1986) Run probabilities and the motion of a particle on a given path. J. Appl. Prob. 23, 28–41.CrossRefGoogle Scholar
Saperstein, B. (1973) On the occurrence of n successes within N Bernoulli trials. Technometrics 15, 809–818.Google Scholar
Schwager, S. J. (1983) Run probabilities in sequences of Markov-dependent trials. J. Amer. Statist. Assoc. 78, 168–175.CrossRefGoogle Scholar
Solov'Eov, A. D. (1966) A combinatorial identity and its application to the problem about the first occurrence of a rare event. Teor. Veroyatnost. i Primenen. 11, 313–320.Google Scholar