## The Annals of Mathematical Statistics

### Statistical Methods in Markov Chains

Patrick Billingsley

#### Abstract

This paper is an expository survey of the mathematical aspects of statistical inference as it applies to finite Markov chains, the problem being to draw inferences about the transition probabilities from one long, unbroken observation $\{x_1, x_2, \cdots, x_n\}$ on the chain. The topics covered include Whittle's formula, chi-square and maximum-likelihood methods, estimation of parameters, and multiple Markov chains. At the end of the paper it is briefly indicated how these methods can be applied to a process with an arbitrary state space or a continuous time parameter. Section 2 contains a simple proof of Whittle's formula; Section 3 provides an elementary and self-contained development of the limit theory required for the application of chi-square methods to finite chains. In the remainder of the paper, the results are accompanied by references to the literature, rather than by complete proofs. As is usual in a review paper, the emphasis reflects the author's interests. Other general accounts of statistical inference on Markov processes will be found in Grenander [53], Bartlett [9] and [10], Fortet [35], and in my monograph [18]. I would like to thank Paul Meier for a number of very helpful discussions on the topics treated in this paper, particularly those of Section 3.

#### Article information

Source
Ann. Math. Statist., Volume 32, Number 1 (1961), 12-40.

Dates
First available in Project Euclid: 27 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aoms/1177705136

Digital Object Identifier
doi:10.1214/aoms/1177705136

Mathematical Reviews number (MathSciNet)
MR123420

Zentralblatt MATH identifier
0104.12802

JSTOR