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

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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.

MSC

60J05
60F10
60F15

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

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Supported by DFG.