Transient phenomena for Markov chains and applications

We consider a family of irreducible, ergodic and aperiodic Markov chains X(ε) = {X(ε) n, n ≧0} depending on a parameter ε > 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 ε↓ 0. Under assumptions involving at most the existence of moments of order 2 + γ 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.