In , Anderson and Goodman investigated statistical inference about Markov chains for random samples of independently, identically distributed chains, in  Goodman obtained results for single chains and in  Baum and Petrie studied inferences for probabilistic functions of Markov chains. Most of their results are either for a large number of identical chains or for very long chains. In this paper, we shall be mainly concerned with the distributions of certain probabilistic functions of a finite chain and the admissibility of these functions as test statistics. Some similar and related results in this respect, especially those for a two state Markov chain may be found in the works of many authors, such as Fisz , chapter 11, Goodman , Lehmann , pp. 115-6, and Mood . The results in Goodman  dealt more generally with $s$ state Markov chain with $s \geqq 2$ and certain extensions and supplementary results can be found in  and . Throughout this paper, we shall assume that our Markov chain is defined on a finite state space and has a discrete time parameter which takes on nonnegative integers. Our derivations make use of a vector representation of an arbitrary Markov chain as given in Section 2 below. In Section 3, a special sequence of probabilistic functions of a Markov chain is defined and shown to be independently, identically distributed multinomial random vectors under certain assumptions about the transition probabilities. The admissibility of test statistics based on some of these functions is shown in Section 4 and a few examples are given in Section 5.
Chia Kuei Tsao. "Admissibility and Distribution of Some Probabilistic Functions of Discrete Finite State Markov Chains." Ann. Math. Statist. 39 (5) 1646 - 1653, October, 1968. https://doi.org/10.1214/aoms/1177698146