Convergence rates for reversible Markov chains without the assumption of nonnegative definite matrices

Explicit convergence rates in geometric and strong ergodicity for denumerable discrete time Markov chains with general reversible transition matrices are obtained in terms of the geometric moments or uniform moments of the hitting times to a fixed point.Another way by Lyapunov's drift conditions is also used to derive these convergence rates.As a typical example,the discrete time birth-death process(random walk) is studied and the explicit criteria for geometric ergodicity are presented.