- Markov chain, Invariant distribution, Lamperti problem, Asymptotically zero drift, Test (Lyapunov) function, Regularly varying tail behaviour, Convergence to Gamma-distribution, Renewal function, Harmonic function, Non-exponential change of measure, Martingale technique, STOCHASTIC DIFFERENCE-EQUATIONS, LIMIT-THEOREMS, DISTRIBUTIONS, BEHAVIOR, MODELS

Research output: Contribution to journal › Journal article › peer-review

Published

**Potential analysis for positive recurrent Markov chains with asymptotically zero drift : power-type asymptotics.** / Denisov, Denis; Korshunov, Dmitry; Wachtel, Vitali.

Research output: Contribution to journal › Journal article › peer-review

Denisov, D, Korshunov, D & Wachtel, V 2013, 'Potential analysis for positive recurrent Markov chains with asymptotically zero drift: power-type asymptotics', *Stochastic Processes and their Applications*, vol. 123, no. 8, pp. 3027-3051. https://doi.org/10.1016/j.spa.2013.04.011

Denisov, D., Korshunov, D., & Wachtel, V. (2013). Potential analysis for positive recurrent Markov chains with asymptotically zero drift: power-type asymptotics. *Stochastic Processes and their Applications*, *123*(8), 3027-3051. https://doi.org/10.1016/j.spa.2013.04.011

Denisov D, Korshunov D, Wachtel V. Potential analysis for positive recurrent Markov chains with asymptotically zero drift: power-type asymptotics. Stochastic Processes and their Applications. 2013 Aug;123(8):3027-3051. https://doi.org/10.1016/j.spa.2013.04.011

@article{f6f94a525556491499a297cd021089f7,

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

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.",

keywords = "Markov chain, Invariant distribution, Lamperti problem, Asymptotically zero drift, Test (Lyapunov) function, Regularly varying tail behaviour, Convergence to Gamma-distribution, Renewal function, Harmonic function, Non-exponential change of measure, Martingale technique, STOCHASTIC DIFFERENCE-EQUATIONS, LIMIT-THEOREMS, DISTRIBUTIONS, BEHAVIOR, MODELS",

author = "Denis Denisov and Dmitry Korshunov and Vitali Wachtel",

year = "2013",

month = aug,

doi = "10.1016/j.spa.2013.04.011",

language = "English",

volume = "123",

pages = "3027--3051",

journal = "Stochastic Processes and their Applications",

issn = "0304-4149",

publisher = "Elsevier",

number = "8",

}

TY - JOUR

T1 - Potential analysis for positive recurrent Markov chains with asymptotically zero drift

T2 - power-type asymptotics

AU - Denisov, Denis

AU - Korshunov, Dmitry

AU - Wachtel, Vitali

PY - 2013/8

Y1 - 2013/8

N2 - 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.

AB - 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.

KW - Markov chain

KW - Invariant distribution

KW - Lamperti problem

KW - Asymptotically zero drift

KW - Test (Lyapunov) function

KW - Regularly varying tail behaviour

KW - Convergence to Gamma-distribution

KW - Renewal function

KW - Harmonic function

KW - Non-exponential change of measure

KW - Martingale technique

KW - STOCHASTIC DIFFERENCE-EQUATIONS

KW - LIMIT-THEOREMS

KW - DISTRIBUTIONS

KW - BEHAVIOR

KW - MODELS

U2 - 10.1016/j.spa.2013.04.011

DO - 10.1016/j.spa.2013.04.011

M3 - Journal article

VL - 123

SP - 3027

EP - 3051

JO - Stochastic Processes and their Applications

JF - Stochastic Processes and their Applications

SN - 0304-4149

IS - 8

ER -