Abstract:
We study the ergodic properties of Markov chains with an arbitrary
state space and prove a geometric ergodic theorem. The method of the
proof is new: it may be described as an operator method. Our main
result is an ergodic theorem for Harris–Markov chains in the case when
the return time to some fixed set has finite expectation. Our conditions
for the transition function are more general than those used
by Athreya–Ney and Nummelin. Unlike them, we impose restrictions not
on the original transition function but on the transition function of an
embedded Markov chain constructed from the return times to the fixed
set mentioned above. The proof uses the spectral theory of linear
operators on a Banach space.