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 Izv. RAN. Ser. Mat., 2007, Volume 71, Issue 4, Pages 69–102 (Mi izv690)

Occupation times and exact asymptotics of small deviations of Bessel processes for $L^p$-norms with $p>0$

V. R. Fatalov

M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

Abstract: We prove theorems on exact asymptotics of the distributions of integral functionals of the occupation time of Bessel processes. Using these results, we obtain exact asymptotics of small deviations for Bessel processes in the $L^p$-norm. We use Laplace's method for the occupation times of Markov processes with continuous time. Computations are carried out for $p=2$ and $p=1$. We also solve extremal problems for the action functional.

DOI: https://doi.org/10.4213/im690

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English version:
Izvestiya: Mathematics, 2007, 71:4, 721–752

Bibliographic databases:

UDC: 519.2
MSC: 60B11, 60B12, 60F05, 60G15, 60G17, 60G60
Revised: 13.04.2006

Citation: V. R. Fatalov, “Occupation times and exact asymptotics of small deviations of Bessel processes for $L^p$-norms with $p>0$”, Izv. RAN. Ser. Mat., 71:4 (2007), 69–102; Izv. Math., 71:4 (2007), 721–752

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/izv690
• https://doi.org/10.4213/im690
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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. V. R. Fatalov, “Exact Asymptotics of Small Deviations for a Stationary Ornstein–Uhlenbeck Process and Some Gaussian Diffusion Processes in the $L_p$-Norm, $2\le p\le\infty$”, Problems Inform. Transmission, 44:2 (2008), 138–155
2. V. R. Fatalov, “Exact asymptotics of Laplace-type Wiener integrals for $L^p$-functionals”, Izv. Math., 74:1 (2010), 189–216
3. V. R. Fatalov, “Small deviations for two classes of Gaussian stationary processes and $L^p$-functionals, $0<p\le\infty$”, Problems Inform. Transmission, 46:1 (2010), 62–85
4. V. R. Fatalov, “Large deviations for distributions of sums of random variables: Markov chain method”, Problems Inform. Transmission, 46:2 (2010), 160–183
5. V. R. Fatalov, “Exact asymptotics of probabilities of large deviations for Markov chains: the Laplace method”, Izv. Math., 75:4 (2011), 837–868
6. Ya. Yu. Nikitin, R. S. Pusev, “The exact asymptotic of small deviations for a series of Brownian functionals”, Theory Probab. Appl., 57:1 (2013), 60–81
7. V. R. Fatalov, “Ergodic means for large values of $T$ and exact asymptotics of small deviations for a multi-dimensional Wiener process”, Izv. Math., 77:6 (2013), 1224–1259
8. V. R. Fatalov, “Integrals of Bessel processes and multi-dimensional Ornstein–Uhlenbeck processes: exact asymptotics for $L^p$-functionals”, Izv. Math., 82:2 (2018), 377–406
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