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Journal of Theoretical Probability

, Volume 32, Issue 2, pp 934–949 | Cite as

Small Ball Probabilities for Certain Gaussian Random Fields

  • Leonid V. RozovskyEmail author
Article

Abstract

Our main goal is to study the behavior of the tail probabilities \({\mathbf{P}}(V^2<r)\) as \(r\rightarrow 0\), where \(V^2\) is defined by the following double sum
$$\begin{aligned} V^2 =\pi ^{-4}\,\sum \limits _{i,j\ge 1} \Big ((i+b)\,(j + {\delta })\Big )^{-2}\,\xi _{ij}^2, \end{aligned}$$
where \( \{\xi _{ij}\} \) are independent standard normal random variables, and b and \({\delta }\) are constants: \(b>-1\) and \(\delta >-1\).

Keywords

Small deviations Karhunen–Loève expansion Gaussian random field Tensor product \( L_2\)-norm 

Mathematics Subject Classification (2010)

60G50 60G15 

Notes

Acknowledgements

The author is indebted to professors M. A. Lifshits, Ya. Yu. Nikitin and A. I. Nazarov (Saint Petersburg State University) for helpful discussions, and an anonymous referee for valuable comments.

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2017

Authors and Affiliations

  1. 1.Department of MathematicsSt. Petersburg Chemical Pharmaceutical AcademySt. PetersburgRussia

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