Approximate probabilities for runs and patterns in i.i.d. and Markov-dependent multistate trials
James C. Fu and Brad C. Johnson
Source: Adv. in Appl. Probab.
Volume 41, Number 1
(2009), 292-308.
Abstract
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.
Primary Subjects: 60E05
Secondary Subjects: 60J10
Keywords: Finite Markov chain imbedding; rate functions; multistate trial
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Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.aap/1240319586
Digital Object Identifier: doi:10.1239/aap/1240319586
Zentralblatt MATH identifier:
05551344
Mathematical Reviews number (MathSciNet):
MR2514955
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