## The Annals of Mathematical Statistics

### Limit Theorems for Sums of Random Variables Defined on Finite Inhomogeneous Markov Chains

Harry Cohn

#### Abstract

Let $(\Omega, \mathscr{F}, P)$ be a probability space, $\{X_n: n \geqq 1\}$ an inhomogeneous Markov chain assuming a finite number of states defined on this space, $E = \{a_1, \cdots, a_s\}$ the set of its states, $p_j^{(n)} = P\{X_n = a_j\}, p^{(k, n)}_{ij} = P\{X_n = a_j\mid X_k = a_i\}$ for $n = 2,3, \cdots, n > k, a_i, a_j \in E, \{f_n: n \geqq 1\}$ a sequence of real valued functions defined on $E$ and $S_n = f_1(X_1) + \cdots + f_n(X_n)$. To study the Markov chains which are not subjected to "asymptotic independent" restrictions, the author proposes the coefficients $a_{k,n} = \max'_{i\in\{1, \cdots, s\}} \sum^s_{j=1} (pj^{(n)} - p^{(k,n)}_{ij})^+ (n = 2,3, \cdots, n > k)$ where the dash indicates that the max is taken over those $i$ such that $p_i^{(k)} > 0$. Some limit properties of the sums $\{S_n: n \geqq 1\}$ suitably normed, as the behavior of the series of random variables and the strong law of large numbers are investigated. In the end some examples are given and it is proved that the arbitrary homogeneous Markov chains satisfy most of the conditions imposed in the paper.

#### Article information

Source
Ann. Math. Statist., Volume 43, Number 4 (1972), 1283-1292.

Dates
First available in Project Euclid: 27 April 2007

https://projecteuclid.org/euclid.aoms/1177692479

Digital Object Identifier
doi:10.1214/aoms/1177692479

Mathematical Reviews number (MathSciNet)
MR317411

Zentralblatt MATH identifier
0247.60023

JSTOR