Pei‐de Chen1, Richard L. Tweedie1
1Department of Statistics, Colorado State University,Fort Collins, Colorado 80523, United States of America
Tóm tắt
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 (xα)
such that the measures Kθ(xα,
·)[colone ](1−θ)ΣPn(xα,
·)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.
