Asymptotic rate of a Markov process

Abstract

For a Markov process in discrete time, both (a) the time-dependent behaviour of the process at time t, and (b) the behaviour of moments (if finite) of the process, whether or not stationary, as a function of time-lag t, are determined by solutions to a single equation, the key equation. The state space of the process is required only to be both a measure space, and a locally convex topological vector space, which may have infinite dimension. The asymptotic behaviour of both (a) and (b), for large times t, is shown to be (nearly) the same, being proportional to tξe−βt, where the exponent β is calculable from an eigenvalue of a linear operator, describing the Markov process. For a large class of Markov processes, the spectrum of this operator is shown to consist of discrete eigenvalues only. Queueing applications are discussed.