Transient phenomena for Markov chains and applications

We consider a family of irreducible, ergodic and aperiodic Markov chains X(.) = {X(.)n, n .0} depending on a parameter . > 0, so that the local drifts have a critical behaviour (in terms of Pakes' lemma). The purpose is to analyse the steady-state distributions of these chains (in the sense of weak convergence), when .. 0. Under assumptions involving at most the existence of moments of order 2 + . for the jumps, we show that, whenever X(0) is not ergodic, it is possible to characterize accurately these limit distributions. Connections with the gamma and uniform distributions are revealed. An application to the well-known ALOHA network is given.