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Asymptotic behaviour of random tridiagonal Markov chains in biological applications
Discrete-time discrete-state random Markov chains with a tridiagonal
generator are shown to have a random attractor consisting of singleton subsets,
essentially a random path, in the simplex of probability vectors. The proof
uses the Hilbert projection metric and the fact that the linear cocycle
generated by the Markov chain is a uniformly contractive mapping of the
positive cone into itself. The proof does not involve probabilistic properties
of the sample path and is thus equally valid in the nonautonomous deterministic
context of Markov chains with, say, periodically varying transitions
probabilities, in which case the attractor is a periodic path.Comment: 13 pages, 22 bibliography references, submitted to DCDS-B, added
references and minor correction