Ergodic Theory of Markov Chains Admitting an Infinite Invariant Measure

Ergodic Theory of Markov Chains Admitting an Infinite Invariant Measure.

Countable Infinitary Theories Admitting an Invariant Measure

Let L be a countable language. We characterize, in terms of definable closure, those countable theories Σ of Lω1,ω(L) for which there exists an S∞-invariant probability measure on the collection of models of Σ with underlying set N. Restricting to Lω,ω(L), this answers an open question of Gaifman from 1964, via a translation between S∞-invariant measures and Gaifman’s symmetric measure-models w...

A Perturbation Theory for Ergodic Properties of Markov Chains

Perturbations to Markov chains and Markov processes are considered. The unperturbed problem is assumed to be geometrically er-godic in the sense usually established through use of Foster-Lyapunov drift conditions. The perturbations are assumed to be uniform, in a weak sense, on bounded time intervals. The long-time behaviour of the perturbed chain is studied. Applications are given to numerical...

Ergodic Theory for Countable State Space Markov Chains

Here we want to see to what extent the results from Doeblin’s theory for finite state spaces extend to the case when the state space is countably infinite. Throughout we will use π to denote the row vector whose jth component is πj = ( E[ρj |X0 = j] )−1. Theorem (A). If j is transient, then πj = 0 = limn→∞(P)ij for all i. Proof. Because P(ρj = ∞|X0 = j) > 0, it is clear that πj = 0. Thus, all t...

Generalized 3x + 1 Mappings: Markov Chains and Ergodic Theory

This paper reviews connections of the 3x + 1 mapping and its generalizations with ergodic theory and Markov chains. The work arose out of efforts to describe the observed limiting frequencies of divergent trajectories in the congruence classes (mod m).