## The Annals of Mathematical Statistics

- Ann. Math. Statist.
- Volume 42, Number 6 (1971), 1992-2007.

### Asymptotically Optimal Tests for Finite Markov Chains

#### Abstract

A discrete time, finite Markov chain with fixed initial state and stationary transition behavior is considered. Using Whittle's formula a large deviation result (similar to Hoeffding's result for one multinomial distribution) is obtained for the transition count matrix of a path of the chain of arbitrary length. This result is then used in the asymptotic comparison of a given sequence of tests about the transition probability matrix with a suitably constructed sequence of likelihood ratio tests. It is assumed that the sizes of these tests decrease to zero at a certain rate as the length of the observed path increases. The comparison is carried out at fixed alternatives in terms of the behavior of the ratio of type-II-error probabilities.

#### Article information

**Source**

Ann. Math. Statist., Volume 42, Number 6 (1971), 1992-2007.

**Dates**

First available in Project Euclid: 27 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aoms/1177693067

**Digital Object Identifier**

doi:10.1214/aoms/1177693067

**Mathematical Reviews number (MathSciNet)**

MR300383

**Zentralblatt MATH identifier**

0246.62085

**JSTOR**

links.jstor.org

#### Citation

Boza, Luis B. Asymptotically Optimal Tests for Finite Markov Chains. Ann. Math. Statist. 42 (1971), no. 6, 1992--2007. doi:10.1214/aoms/1177693067. https://projecteuclid.org/euclid.aoms/1177693067