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Limit theorems for the number of occurrences of consecutive k successes in n Markovian trials

Part of: Markov processes Limit theorems

Published online by Cambridge University Press:  14 July 2016

Y. H. Wang  and
Shuixin Ji
Y. H. Wang*
Affiliation:
Concordia University
Shuixin Ji*
Affiliation:
Concordia University
*
∗Postal address: Department of Mathematics and Statistics, Concordia University, 1455 de Maisonneuve Blvd W, Montréal, PQ Canada H3G 1M8.
∗Postal address: Department of Mathematics and Statistics, Concordia University, 1455 de Maisonneuve Blvd W, Montréal, PQ Canada H3G 1M8.
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Abstract

We present a method of deriving the limiting distributions of the number of occurrences of success (S) runs of length k for all types of runs under the Markovian structure with stationary transition probabilities. In particular, we consider the following four bestknown types. 1. A string of S of exact length k preceded and followed by an F, except the first run which may not be preceded by an F, or the last run which may not be followed by an F. 2. A string of S of length k or more. 3. A string of S of exact length k, where recounting starts immediately after a run occurs. 4. A string of S of exact length k, allowing overlapping runs. It is shown that the limits are convolutions of two or more distributions with one of them being either Poisson or compound Poisson, depending on the type of runs in question. The completely stationary Markov case and the i.i.d. case are also treated.

Keywords

RUNSMARKOV CHAINSSTATIONARY TRANSITION PROBABILITIESCONVERGENCEPOISSONCOMPOUND POISSONRELIABILITYCONSECUTIVE-k-OUT-OF n: F SYSTEM

MSC classification

Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Secondary: 60F05: Central limit and other weak theorems
Type
Research Papers
Information
Journal of Applied Probability , Volume 32 , Issue 3 , September 1995 , pp. 727 - 735
Copyright
Copyright © Applied Probability Trust 1995 

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Footnotes

The first author's research is partially supported by the Natural Sciences and Engineering Council of Canada, #OGPIN 014.

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