Vol. 111, No. 2, 1984

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Poisson process over σ-finite Markov chains

Guillermo Grabinsky

Vol. 111 (1984), No. 2, 301–315

There is a well-known construction which associates with each σ-finite measure space (X,𝒮) a certain stochastic process {N(F) : F ∈𝒮(F) < ∞} called the Poisson process over (X,𝒮). Any μ-preserving bimeasurable map τ on X “lifts” to a probability preserving map T, characterized by N(F) T = N(τ1F). We show the following: If τ is the shift arising from a Markov chain preserving a σ-finite measure with stochastic matrix (pi,j)i,jN. Then T is a Bernoulli shift iff pi,jn 0i,j N as n →∞. If, in addition, τ has a recurrent state or if it is transient and (𝒮) is not completely atomic, then T has infinite entropy. The analogous results are valid for ν-step Markov chains preserving a σ-finite measure (ν > 1).

Mathematical Subject Classification 2000
Primary: 28D05
Secondary: 60J10
Received: 14 July 1982
Published: 1 April 1984
Guillermo Grabinsky