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

Denisov, Denis and Korshunov, Dmitry and Wachtel, Vitali (2013) Potential analysis for positive recurrent Markov chains with asymptotically zero drift:power-type asymptotics. Stochastic Processes and their Applications, 123 (8). pp. 3027-3051. ISSN 0304-4149

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Abstract

We consider a positive recurrent Markov chain on R+ with asymptotically zero drift which behaves like -c(1)/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 c(2)/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.

Item Type:
Journal Article
Journal or Publication Title:
Stochastic Processes and their Applications
Uncontrolled Keywords:
/dk/atira/pure/subjectarea/asjc/2600/2613
Subjects:
ID Code:
74011
Deposited By:
Deposited On:
18 Jun 2015 05:58
Refereed?:
Yes
Published?:
Published
Last Modified:
01 Jan 2020 09:15