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Augmented truncations of infinite stochastic matrices. (English) Zbl 0628.15019
A stochastic matrix P=(p ij ) is said to be a Markov matrix if there exists a column j 0 and a real number ϵ>0 so that p ij 0 >ϵ for all i. A stochastic matrix P=(p ij ) is said to be upper-Hessenberg (lower-Hessenberg) if p ij =0 when i>j+1 (j>i+1). For a stochastic P let π denote a stationary distribution for P. Let (n) P ˜ be an n×n stationary matrix so that (n) P ˜ (n) P elementwise. Let (n) π denote a stationary distribution for (n) P ˜. It is shown that if P is Markov, then (n) π is unique for sufficiently large n and (n) ππ as n. 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 (n) P. An example is given to show that the result need not hold if these restrictions are not met.
Reviewer: R.Sinkhorn
15A52Random matrices (MSC2000)
15A51Stochastic matrices (MSC2000)
60J10Markov chains (discrete-time Markov processes on discrete state spaces)