Stefano Isola
On the rate of convergence to equilibrium
for countable ergodic Markov chains
(49K, Plain-Tex)
ABSTRACT. Using elementary methods, we prove
that for a countable Markov
chain $P$ of ergodic degree $\ell \geq 1$ the rate of convergence
towards the stationary distribution is $o(n^{-(\ell-1)})$,
provided the initial distribution satisfies certain
conditions of asymptotic decay. An example, modelling
temporal intermittency in dynamical system theory,
is worked out
in detail, illustrating the relationships between convergence
behaviour, analytic properties of the generating functions
associated to transition probabilities
and spectral properties of the Markov operator
$P$ on the Banach space
$\ell_1$. Explicit conditions allowing to obtain
either optimal bounds or
the actual asymptotics for the rate of
convergence are also discussed.