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

• Published in: Stochastic processes and their applications Vol. 123; no. 8; pp. 3027 - 3051
• Main Authors: Denisov, Denis, Korshunov, Dmitry, Wachtel, Vitali
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 01.08.2013
• Subjects: Asymptotically zero drift; Convergence to [formula omitted]-distribution; Harmonic function; Invariant distribution; Lamperti problem; Markov chain; Martingale technique; Non-exponential change of measure; Regularly varying tail behaviour; Renewal function; Test (Lyapunov) function; Convergence to Γ-distribution; Martingale technique; 60F10; Asymptotically zero drift; Test (Lyapunov) function; Harmonic function; Non-exponential change of measure; Regularly varying tail behaviour; Markov chain; Lamperti problem; 60J05; Renewal function; 60F15; Invariant distribution
• ISSN: 0304-4149; 1879-209X
• DOI: 10.1016/j.spa.2013.04.011

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.