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October, 1990 On Products of Nonnegative Matrices
Harry Cohn, Olle Nerman
Ann. Probab. 18(4): 1806-1815 (October, 1990). DOI: 10.1214/aop/1176990650


A representation for products of finite nonnegative matrices is given in terms of products of stochastic matrices and as a result Markov chain arguments are used to derive ratio limit properties. In particular, we obtain necessary and sufficient conditions for weak ergodicity and give a probabilistic proof of the Coale-Lopez theorem. In the general case, there are several sequences of sets of partitions of the state space corresponding to an associated nonhomogeneous Markov chain which lead to a number of ratio product limits. Asymptotic column proportionality, characteristic of weak ergodicity, may occur only inside each sequence of sets with one possible exception.


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Harry Cohn. Olle Nerman. "On Products of Nonnegative Matrices." Ann. Probab. 18 (4) 1806 - 1815, October, 1990.


Published: October, 1990
First available in Project Euclid: 19 April 2007

zbMATH: 0719.60037
MathSciNet: MR1071827
Digital Object Identifier: 10.1214/aop/1176990650

Primary: 15A48
Secondary: 60F99 , 60J05 , 60J45

Keywords: atomic set , Harmonic function , Markov chain , Nonnegative matrix , ratio limit , space-time chain , stochastic matrix , tail $\sigma$-field , weak ergodicity

Rights: Copyright © 1990 Institute of Mathematical Statistics


Vol.18 • No. 4 • October, 1990
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