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Theory of Probability and Mathematical Statistics
Theory of Probability and Mathematical Statistics
ISSN 1547-7363(online) ISSN 0094-9000(print)

 

The invariance principle for the Ornstein-Uhlenbeck process with fast Poisson time: An estimate for the rate of convergence


Authors: B. V. Bondarev and A. V. Baev
Translated by: S. Kvasko
Original publication: Teoriya Imovirnostei ta Matematichna Statistika, tom 76 (2007).
Journal: Theor. Probability and Math. Statist. 76 (2008), 15-22
MSC (2000): Primary 60E15, 60H10; Secondary 60F17
Published electronically: July 10, 2008
MathSciNet review: 2368735
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Abstract | References | Similar Articles | Additional Information

Abstract: We consider the invariance principle for

$\displaystyle \varsigma _n (t) = n^{ - 1/2} \int_0^{Z(nt)} \xi (s)\,ds, $

where $ \xi (s)$ is the Ornstein-Uhlenbeck process and $ Z(t)$, $ t \geq 0$, is the Poisson process such that $ {\mathsf E} Z(t) = \lambda (t)$. We prove that

$\displaystyle {\mathsf P}\left\{\sup_{0 \leq t \leq T} \left\vert {\varsigma _n... ...ma\gamma n^{ - 1/2} W(\lambda (nt))} \right\vert >r_n \right\} \leq \alpha _n, $

where $ r_n\to 0$ and $ \alpha _n \to 0$ as $ n \to+\infty$.


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Additional Information

B. V. Bondarev
Affiliation: Department of Probability Theory and Mathematical Statistics, Faculty for Mathematics, Donetsk National University, Universiets’ka Street, 24, 83055 Donetsk, Ukraine
Email: bvbondarev@cable.netlux.org

A. V. Baev
Affiliation: Department of Probability Theory and Mathematical Statistics, Faculty for Mathematics, Donetsk National University, Universiets’ka Street, 24, 83055 Donetsk, Ukraine
Email: tv@matfak.dongu.donetsk.ua

DOI: http://dx.doi.org/10.1090/S0094-9000-08-00727-8
PII: S 0094-9000(08)00727-8
Keywords: Ornstein--Uhlenbeck process, distribution of the supremum, Poisson process
Received by editor(s): January 6, 2006
Published electronically: July 10, 2008
Article copyright: © Copyright 2008 American Mathematical Society