Poisson process over σ-finite Markov chains

Guillermo Grabinsky

Vol. 111 (1984), No. 2, 301–315

DOI: 10.2140/pjm.1984.111.301

Abstract

There is a well-known construction which associates with each σ-finite measure space (X,𝒮,μ) a certain stochastic process {N(F) : F ∈𝒮,μ(F) < ∞} called the Poisson process over (X,𝒮,μ). Any μ-preserving bimeasurable map τ on X “lifts” to a probability preserving map T, characterized by N(F) ∘ T = N(τ⁻¹F). We show the following: If τ is the shift arising from a Markov chain preserving a σ-finite measure with stochastic matrix (p_i,j)_i,j∈N. Then T is a Bernoulli shift iff p_i,jⁿ → 0∀i,j ∈ N as n →∞. If, in addition, τ has a recurrent state or if it is transient and (𝒮,μ) 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

Milestones

Received: 14 July 1982

Published: 1 April 1984