Potential analysis for positive recurrent Markov chains with asymptotically zero drift: Power-type asymptotics

Denis Denisov, Dmitry Korshunov and Vitali Wachtel

Stochastic Processes and their Applications, 2013, vol. 123, issue 8, 3027-3051

Abstract: We consider a positive recurrent Markov chain on R+ with asymptotically zero drift which behaves like −c1/x at infinity; this model was first considered by Lamperti. We are interested in tail asymptotics for the stationary measure. Our analysis is based on construction of a harmonic function which turns out to be regularly varying at infinity. This harmonic function allows us to perform non-exponential change of measure. Under this new measure Markov chain is transient with drift like c2/x at infinity and we compute the asymptotics for its Green function. Applying further the inverse transform of measure we deduce a power-like asymptotic behaviour of the stationary tail distribution. Such a heavy-tailed stationary measure happens even if the jumps of the chain are bounded. This model provides an example where possibly bounded input distributions produce non-exponential output.

Keywords: Markov chain; Invariant distribution; Lamperti problem; Asymptotically zero drift; Test (Lyapunov) function; Regularly varying tail behaviour; Convergence to Γ-distribution; Renewal function; Harmonic function; Non-exponential change of measure; Martingale technique (search for similar items in EconPapers)
Date: 2013
References: View complete reference list from CitEc
Citations: View citations in EconPapers (1)

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DOI: 10.1016/j.spa.2013.04.011

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