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

Research output: Contribution to Journal/MagazineJournal articlepeer-review

<mark>Journal publication date</mark>08/2013
<mark>Journal</mark>Stochastic Processes and their Applications
Issue number8
Number of pages25
Pages (from-to)3027-3051
Publication StatusPublished
Early online date17/04/13
<mark>Original language</mark>English


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.