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Augmented truncations of infinite stochastic matrices. (English) Zbl 0628.15019
A stochastic matrix $P=\left({p}_{ij}\right)$ is said to be a Markov matrix if there exists a column ${j}_{0}$ and a real number $ϵ>0$ so that ${p}_{i{j}_{0}}>ϵ$ for all i. A stochastic matrix $P=\left({p}_{ij}\right)$ is said to be upper-Hessenberg (lower-Hessenberg) if ${p}_{ij}=0$ when $i>j+1$ $\left(j>i+1\right)$. For a stochastic P let $\pi$ denote a stationary distribution for P. Let ${}_{\left(n\right)}\stackrel{˜}{P}$ be an $n×n$ stationary matrix so that ${}_{\left(n\right)}\stackrel{˜}{P}{\ge }_{\left(n\right)}P$ elementwise. Let ${}_{\left(n\right)}\pi$ denote a stationary distribution for ${}_{\left(n\right)}\stackrel{˜}{P}$. It is shown that if P is Markov, then ${}_{\left(n\right)}\pi$ is unique for sufficiently large n and ${}_{\left(n\right)}\pi \to \pi$ 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 ${}_{\left(n\right)}P$. An example is given to show that the result need not hold if these restrictions are not met.
Reviewer: R.Sinkhorn
##### MSC:
 15A52 Random matrices (MSC2000) 15A51 Stochastic matrices (MSC2000) 60J10 Markov chains (discrete-time Markov processes on discrete state spaces)