## Advances in Applied Probability

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

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

James C. Fu and Brad C. Johnson

#### Abstract

Let *X*_{n}(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{*X*_{n}(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

**Source**

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

**Dates**

First available in Project Euclid: 21 April 2009

**Permanent link to this document**

http://projecteuclid.org/euclid.aap/1240319586

**Digital Object Identifier**

doi:10.1239/aap/1240319586

**Mathematical Reviews number (MathSciNet)**

MR2514955

**Zentralblatt MATH identifier**

1166.60008

**Subjects**

Primary: 60E05: Distributions: general theory

Secondary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)

**Keywords**

Finite Markov chain imbedding rate functions multistate trial

#### Citation

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. http://projecteuclid.org/euclid.aap/1240319586.