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Run probabilities and the motion of a particle on a given path

Published online by Cambridge University Press:  14 July 2016

A. Reza Soltani  and
A. Khodadadi
A. Reza Soltani*
Affiliation:
Shiraz University
A. Khodadadi*
Affiliation:
Shiraz University
*
∗Postal address: Department of Mathematics and Statistics, College of Arts and Sciences, Shiraz University, Shiraz, Iran.
∗Postal address: Department of Mathematics and Statistics, College of Arts and Sciences, Shiraz University, Shiraz, Iran.
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Abstract

Let {Xn} be a sequence of independent (or Markov dependent) trials taking values in a given set S. Let JR be a given path of length k in S, i.e. R is a run of length k whose elements come from S. {Xn} may indicate the motion of a particle on S. We consider the problem of finding the probability that at trial m, the particle has for the first time moved length l ≦ k on R which is equivalent to finding the probability of the first occurrence of any subrun of length l ≦ k of R. In the case of l = k this gives the result of Schwager [6].

Keywords

LEADING NUMBERSMARKOV PROCESSESSUBRUNS
Type
Research Papers
Information
Journal of Applied Probability , Volume 23 , Issue 1 , March 1986 , pp. 28 - 41
Copyright
Copyright © Applied Probability Trust 1986 

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References

[1] Bradley, J. V. (1968) Distribution-Free Statistical Tests. Prentice-Hall, Englewood Cliffs, NJ.Google Scholar
[2] Feller, W. (1968) An Introduction to Probability Theory and its Applications. Wiley, New York.Google Scholar
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[6] Schwager, S. J. (1983) Run probabilities in sequences of Markov-dependent trials. J. Amer. Statist. Assoc. 78, 168–175.CrossRefGoogle Scholar