## The Annals of Mathematical Statistics

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

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

#### 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

**Permanent link to this document**

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**

links.jstor.org

#### Citation

Cohn, Harry. Limit Theorems for Sums of Random Variables Defined on Finite Inhomogeneous Markov Chains. Ann. Math. Statist. 43 (1972), no. 4, 1283--1292. doi:10.1214/aoms/1177692479. https://projecteuclid.org/euclid.aoms/1177692479