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August, 1972 Limit Theorems for Sums of Random Variables Defined on Finite Inhomogeneous Markov Chains
Harry Cohn
Ann. Math. Statist. 43(4): 1283-1292 (August, 1972). DOI: 10.1214/aoms/1177692479


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.


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Harry Cohn. "Limit Theorems for Sums of Random Variables Defined on Finite Inhomogeneous Markov Chains." Ann. Math. Statist. 43 (4) 1283 - 1292, August, 1972.


Published: August, 1972
First available in Project Euclid: 27 April 2007

zbMATH: 0247.60023
MathSciNet: MR317411
Digital Object Identifier: 10.1214/aoms/1177692479

Rights: Copyright © 1972 Institute of Mathematical Statistics


Vol.43 • No. 4 • August, 1972
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