Research output: Contribution to Journal/Magazine › Journal article › peer-review
Research output: Contribution to Journal/Magazine › Journal article › peer-review
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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 -