Orthogonal measures and absorbing sets for Markov chains

Abstract

For a general state space Markov chain on a space (X, [Bscr ](X)), the existence of a Doeblin decomposition, implying the state space can be written as a countable union of absorbing ‘recurrent’ sets and a transient set, is known to be a consequence of several different conditions all implying in some way that there is not an uncountable collection of absorbing sets. These include

([Mscr ]) there exists a finite measure which gives positive mass to each absorbing subset of X;

([Gscr ]) there exists no uncountable collection of points (x_α) such that the measures K_θ(x_α, ·)[colone ](1−θ)ΣPⁿ(x_α, ·)θⁿ are mutually singular;

([Cscr ]) there is no uncountable disjoint class of absorbing subsets of X.

We prove that if [Bscr ](X) is countably generated and separated (distinct elements in X can be separated by disjoint measurable sets), then these conditions are equivalent. Other results on the structure of absorbing sets are also developed.