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On Null-Recurrent Markov Chains

Published online by Cambridge University Press:  20 November 2018

John Lamperti
John Lamperti*
Affiliation:
Stanford University
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Throughout this paper, the symbol P = [Pij] will represent the transition probability matrix of an irreducible, null-recurrent Markov process in discrete time. Explanation of this terminology and basic facts about such chains may be found in (6, ch. 15). It is known (3) that for each such matrix P there is a unique (except for a positive scalar multiple) positive vector Q = {qi} such that QP = Q, or

1

this vector is often called the "invariant measure" of the Markov chain.

The first problem to be considered in this paper is that of determining for which vectors U(0) = {μi(0)} the vectors U(n) converge, or are summable, to the invariant measure Q, where U(n) = U(0)Pn has components

2

In § 2, this problem is attacked for general P. The main result is a negative one, and shows how to form U(0) for which U(n) will not be (termwise) Abel summable.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 12 , 1960 , pp. 278 - 288
Copyright
Copyright © Canadian Mathematical Society 1960

References

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