## The Annals of Mathematical Statistics

### Asymptotic Inference in Markov Processes

G. G. Roussas

#### Abstract

Let $(\mathscr{X}, \mathscr{a})$ be a measurable space and $\Theta$ be an open subset of a $k$-dimensional Euclidean space. For each $\theta \varepsilon \Theta$ let $P_\theta$ be a probability measure on $\mathscr{a}$. We assume that for every $\theta \varepsilon \Theta, \{X_n, n \geqq 0\}$ is a Markov process defined on $(\mathscr{X}, \mathscr{a})$ into $(R, \mathscr{B})$, where $(R, \mathscr{B})$ denotes the Borel real line. Furthermore, we denote by $\mathscr{a}_n$ the $\sigma$-fields induced by the random variables $\{X_0, X_1, \cdots, X_n\}$, and by $P_{n, \theta}$ the restriction of $P_\theta$ to $\mathscr{a}_n$. In the present paper we give conditions under which the sequence of families of probability measures $\{P_{n,\theta}, \theta \varepsilon \Theta\}$ has the desirable property of being differentially asymptotically normal. This implies that in some neighborhood of $\theta \varepsilon \Theta, \{P_{n,\theta}, \theta \varepsilon \Theta\}, n \geqq 0$ can be, for certain problems, approximately treated as if they were normal. For a detailed account of the notions involved in this paper the reader is referred to [5], in particular, Section 5. Also in the Appendix of the present paper one can find the definitions of the concepts most frequently used in this work, including that of the differentially asymptotically normal families of distributions. In Section 2 the required notation is introduced and also the assumptions being made throughout the paper are listed. In Section 3 we state the main result, and in the following subsections we give the proof of it in several steps.

#### Article information

Source
Ann. Math. Statist., Volume 36, Number 3 (1965), 978-992.

Dates
First available in Project Euclid: 27 April 2007

https://projecteuclid.org/euclid.aoms/1177700070

Digital Object Identifier
doi:10.1214/aoms/1177700070

Mathematical Reviews number (MathSciNet)
MR179882

Zentralblatt MATH identifier
0138.11601

JSTOR