Intrinsic Convergence Rate of Countable Markov Chains

#### V.A. Malyshev, F.M. Spieksma

1995, v.1, Issue 2, 203-266

ABSTRACT

The exponential convergence rate to stationarity is very sensitive to perturbations of the transition probabilities. This motivates introducing a parameter that is invariant under perturbations within a finite domain, called intrinsic rate''. The intuitive interpretation of this invariant is the exponential rate at which the Markov chain converges back to a finite domain from infinity''. For random walks in $Z_+$ and $Z^2_+$ we will study this invariant using probabilistic methods, in particular Large Deviations techniques, and analytic methods. Thus we connect convergence rates, action functionals, singularities of generating functions and spectral properties of transition matrices.

Keywords: countable Markov chains,exponential convergence,Large Deviations,essential spectrum