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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
Journal: Theor. Probability and Math. Statist. 76 (2008), 15-22
MSC (2000): Primary 60E15, 60H10; Secondary 60F17
DOI: https://doi.org/10.1090/S0094-9000-08-00727-8
Published electronically: July 10, 2008
MathSciNet review: 2368735
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Abstract: We consider the invariance principle for \[ \varsigma _n (t) = n^{ - 1/2} \int _0^{Z(nt)} \xi (s) ds, \] 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 \[ {\mathsf P}\left \{\sup _{0 \leq t \leq T} \left | {\varsigma _n (t) -\frac \sigma \gamma n^{ - 1/2} W(\lambda (nt))} \right | >r_n \right \} \leq \alpha _n, \] 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

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