Strong Gaussian approximation of metastable density-dependent Markov chains on large time scales

Abstract

We consider density-dependent Markov chains converging, as the scale parameter K>0 goes to infinity, to the solution of an ODE admitting an exponentially stable equilibrium point. We provide a new strong approximation of the density by a Gaussian process, based on a construction of Kurtz using the Komlós–Major–Tusnády theorem. We show that given any threshold ɛ(K)≪1 greater than a multiple of log(K)/K, the time the error needs to reach ɛ(K) is at least of order exp(VKɛ(K)), for some V>0. We discuss consequences on moderate deviations, applications to a logistic birth-and-death process conditioned to survive and to an epidemic model.