A Marginal Ergodic Theorem
In recent years there have been several papers giving examples of
Markov Chain Monte Carlo (MCMC) algorithms whose invariant measures
are improper (have infinite mass) and which therefore are not
positive recurrent, yet which have subchains from which valid
inference can be derived. These are nonergodic (not having a
limiting distribution) Markov chains (MC's) that can be written,
possibly after transformation, as Z = {Z(n) = Z(z,n); n >= 0} =
{(X(n),Y(n); n >= 0} for which the subchain X(n) is ergodic
(has a limiting distribution). This paper gives a marginal
ergodic theorem which (a) gives conditions on Z
guaranteeing that the subchain X is ergodic, (b) gives a
formula for computing the limiting distribution in case it exists,
and (c) gives a formula for bounding the liminf and
limsup as n goes to infinity of the distribution of X(n) in
case the limit does not exist.
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