Advances in Applied Probability

Approximate probabilities for runs and patterns in i.i.d. and Markov-dependent multistate trials

James C. Fu and Brad C. Johnson

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Let Xn(Λ) be the number of nonoverlapping occurrences of a simple pattern Λ in a sequence of independent and identically distributed (i.i.d.) multistate trials. For fixed k, the exact tail probability 𝙿{Xn(Λ) < k} is difficult to compute and tends to 0 exponentially as n → ∞. In this paper we use the finite Markov chain imbedding technique and standard matrix theory results to obtain an approximation for this tail probability. The result is extended to compound patterns, Markov-dependent multistate trials, and overlapping occurrences of Λ. Numerical comparisons with Poisson and normal approximations are provided. Results indicate that the proposed approximations perform very well and do significantly better than the Poisson and normal approximations in many cases.

Article information

Adv. in Appl. Probab. Volume 41, Number 1 (2009), 292-308.

First available in Project Euclid: 21 April 2009

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60E05: Distributions: general theory
Secondary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)

Finite Markov chain imbedding rate functions multistate trial


Fu, James C.; Johnson, Brad C. Approximate probabilities for runs and patterns in i.i.d. and Markov-dependent multistate trials. Adv. in Appl. Probab. 41 (2009), no. 1, 292--308. doi:10.1239/aap/1240319586.

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