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2 results

  • 8 - Tail Analysis for Recurrent Markov Chains with Drift Proportional to 1/x

  • Denis Denisov, University of Manchester, Dmitry Korshunov, Lancaster University, Vitali Wachtel, Universität Bielefeld, Germany
  • Book:
    Markov Chains with Asymptotically Zero Drift
    Published online:
    24 April 2025
    Print publication:
    08 May 2025, pp 236-280

  • Summary

    In Chapter 8 we consider a recurrent Markov chain possessing an invariant measure which is either probabilistic in the case of positive recurrence or σ-finite in the case of null recurrence. Our main aim here is to describe the asymptotic behaviour of the invariant distribution tail for a class of Markov chains with asymptotically zero drift proportional to 1/x. We start with a result which states that a typical stationary Markov chain with asymptotically zero drift always generates a heavy-tailed invariant distribution, which is very different from the case of Markov chains with asymptotically negative drift bounded away from zero. Then we develop techniques needed for deriving precise tail asymptotics of power type.

Markov Chains with Asymptotically Zero Drift
  • Lamperti's Problem
  • Denis Denisov, Dmitry Korshunov, Vitali Wachtel
  • Published online:
    24 April 2025
    Print publication:
    08 May 2025
  • This text examines Markov chains whose drift tends to zero at infinity, a topic sometimes labelled as 'Lamperti's problem'. It can be considered a subcategory of random walks, which are helpful in studying stochastic models like branching processes and queueing systems. Drawing on Doob's h-transform and other tools, the authors present novel results and techniques, including a change-of-measure technique for near-critical Markov chains. The final chapter presents a range of applications where these special types of Markov chains occur naturally, featuring a new risk process with surplus-dependent premium rate. This will be a valuable resource for researchers and graduate students working in probability theory and stochastic processes.