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

Abstract

The main goal of this text is comprehensive study of time homogeneous Markov chains on the real line whose drift tends to zero at infinity, we call such processes Markov chains with asymptotically zero drift. Traditionally this topic is referred to as Lamperti's problem. Time homogeneous Markov chains with asymptotically zero drift may be viewed as a subclass of perturbed in space random walks. The latter are of basic importance in the study of various applied stochastic models, among them branching and risk processes, queueing systems etc. Random walks generated by sums of independent identically distributed random variables are well studied, see e.g. classical textbooks by W. Feller (1971), V.V. Petrov (1975), or F. Spitzer (1964); for the recent development of the theory of random walks we refer to A.A. Borovkov and K.A. Borovkov (2008). There are many monographs devoted to various applications where random walks play a crucial role, let us just mention books on ruin and queueing processes by S. Asmussen (2000, 2003); on insurance and finance by P. Embrechts, C. Kluppelberg, and T. Mikosch (1997), and T. Rolski, H. Schmidli, V. Schmidt, and J. Teugels (1998); and on stochastic difference equations by D. Buraczewski, E. Damek and T. Mikosch (2016). In the same applied stochastic models, if one allows the process considered to be dependent on the current state of the process, we often get a Markov chain which has asymptotically zero drift, we demonstrate that in the last chapter, where we particularly discuss branching and risk processes, stochastic difference equations and ALOHA network.