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

Author

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  • Denisov, Denis
  • Korshunov, Dmitry
  • Wachtel, Vitali

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.

Suggested Citation

  • Denisov, Denis & Korshunov, Dmitry & Wachtel, Vitali, 2013. "Potential analysis for positive recurrent Markov chains with asymptotically zero drift: Power-type asymptotics," Stochastic Processes and their Applications, Elsevier, vol. 123(8), pages 3027-3051.
    • Handle: RePEc:eee:spapps:v:123:y:2013:i:8:p:3027-3051
    DOI: 10.1016/j.spa.2013.04.011
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    Cited by:

    1. Denisov, Denis & Gotthardt, Niklas & Korshunov, Dmitry & Wachtel, Vitali, 2024. "Probabilistic approach to risk processes with level-dependent premium rate," Insurance: Mathematics and Economics, Elsevier, vol. 118(C), pages 142-156.

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