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Stopping Probabilities for Patterns in Markov Chains

Part of: Stochastic processes Markov processes

Published online by Cambridge University Press:  30 January 2018

Renato Jacob Gava  and
Danilo Salotti
Renato Jacob Gava*
Affiliation:
Universidade de São Paulo
Danilo Salotti*
Affiliation:
Fundação Educacional Inaciana Padre Sabóia de Medeiros
*
∗ Postal address: Instituto de Matemática e Estatística, Universidade de São Paulo, Rua do Matão 1010, CEP 05508-090, São Paulo, SP, Brazil. Email address: gava@ime.usp.br
∗∗ Postal address: Centro Universitário da FEI, Av. Humberto de Alencar Castelo Branco 3972, CEP 09850-901, São Bernardo do Campo, SP, Brazil. Email address: dsalotti@fei.edu.br
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Abstract

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Consider a sequence of Markov-dependent trials where each trial produces a letter of a finite alphabet. Given a collection of patterns, we look at this sequence until one of these patterns appears as a run. We show how the method of gambling teams can be employed to compute the probability that a given pattern is the first pattern to occur.

Keywords

Stopping timewaiting timegambling teammartingaleMarkov chainpatternstopping probability

MSC classification

Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Secondary: 60G42: Martingales with discrete parameter
Type
Research Article
Information
Journal of Applied Probability , Volume 51 , Issue 1 , March 2014 , pp. 287 - 292
Copyright
© Applied Probability Trust 

Footnotes

Research supported by FAPESP fellowship 2012/01432-9.

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