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

Full-text PDF Free Access

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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