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Cut-off and hitting times of a sample of Ornstein-Uhlenbeck processes and its average

Part of: Markov processes

Published online by Cambridge University Press:  14 July 2016

B. Lachaud
B. Lachaud*
Affiliation:
Université René Descartes - Paris 5
*
∗Postal address: MAP5, UMR CNRS 8145, Université Paris 5, 45 rue des Saints-Pères, 75270 Paris Cedex 06, France. Email address: beatrice.lachaud@math-info.univ-paris5.fr
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Abstract

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A cut-off phenomenon is shown to occur in a sample of n independent, identically distributed Ornstein-Uhlenbeck processes and its average. Their distributions stay far from equilibrium before a certain O(log(n)) time, and converge exponentially fast after. Precise estimates show that the total variation distance drops from almost 1 to almost 0 over an interval of time of length O(1) around log(n)/(2α), where α is the viscosity coefficient of the sampled process. The distribution of the hitting time of 0 by the average of the sample is computed. As n tends to infinity, the hitting time becomes concentrated around the cut-off instant, and its tails match the estimates given for the total variation distance.

Keywords

Cut-offhitting timeOrnstein-Uhlenbeck process

MSC classification

Primary: 60J60: Diffusion processes
Type
Research Papers
Information
Journal of Applied Probability , Volume 42 , Issue 4 , December 2005 , pp. 1069 - 1080
Copyright
© Applied Probability Trust 2005 

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