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Zbl 0628.15019
Gibson, Diana; Seneta, E.
Augmented truncations of infinite stochastic matrices.
(English)
[J] J. Appl. Probab. 24, 600-608 (1987). ISSN 0021-9002

A stochastic matrix $P=(p\sb{ij})$ is said to be a Markov matrix if there exists a column $j\sb 0$ and a real number $\epsilon >0$ so that $p\sb{ij\sb 0}>\epsilon$ for all i. A stochastic matrix $P=(p\sb{ij})$ is said to be upper-Hessenberg (lower-Hessenberg) if $p\sb{ij}=0$ when $i>j+1$ $(j>i+1)$. For a stochastic P let $\pi$ denote a stationary distribution for P. Let $\sb{(n)}\tilde P$ be an $n\times n$ stationary matrix so that $\sb{(n)}\tilde P\ge\sb{(n)}P$ elementwise. Let $\sb{(n)}\pi$ denote a stationary distribution for $\sb{(n)}\tilde P$. It is shown that if P is Markov, then $\sb{(n)}\pi$ is unique for sufficiently large n and $\sb{(n)}\pi \to \pi$ as $n\to \infty$. The same result holds if P is upper-Hessenberg except that n need not be restricted. The result also holds if P is lower-Hessenberg provided that certain restrictions are placed on the $\sb{(n)}P$. An example is given to show that the result need not hold if these restrictions are not met.
[R.Sinkhorn]
MSC 2000:
*15A52 Random matrices
15A51 Stochastic matrices
60J10 Markov chains with discrete parameter

Keywords: truncations of infinite stochastic matrices; augmentation; last-exit probabilities; stochastic matrix; Markov matrix; upper-Hessenberg; lower- Hessenberg; stationary distribution

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