At the Edge of Criticality: Markov Chains with Asymptotically Zero Drift

Abstract

In Chapter 2 we introduce a classification of Markov chains with asymptotically zero drift, which relies on relations between first and second moments of jumps. We construct an abstract Lyapunov functions which looks similar to functions which characterise the behaviour of diffusions with similar drift and diffusion coefficient. Chapter 3 is devoted to the limiting behaviour of transient chains. Here we prove converges to $\Gamma$ and normal distribution which generalises papers by Lamperti, Kersting and Klebaner. We also determine the asymptotic behaviour of the cumulative renewal function. In Chapter 4 we introduce a general strategy of change of measure for Markov chains with asymptotically zero drift. This is the most important ingredient in our approach to recurrent chains. Chapter 5 is devoted to the study of the limiting behaviour of recurrent chains with the drift proportional to $1/x$. We derive asymptotics for a stationary measure and determine the tail behaviour of recurrence times. All these asymptotics are of power type. In Chapter 6 we show that if the drift is of order $x^{-\beta}$ then moments of all orders $k\le [1/\beta]+1$ are important for the behaviour of stationary distributions and pre-limiting tails. Here we obtain Weibull-like asymptotics. In Chapter 7 we apply our results to different processes, e.g. critical and near-critical branching processes, risk processes with reserve-dependent premium rate, random walks conditioned to stay positive and reflected random walks. In Chapter 8 we consider asymptotically homogeneous in space Markov chains for which we derive exponential tail asymptotics