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 Teor. Veroyatnost. i Primenen.: Year: Volume: Issue: Page: Find

 Teor. Veroyatnost. i Primenen., 2018, Volume 63, Issue 3, Pages 468–481 (Mi tvp5140)

On the exact asymptotics of small deviations of $L_2$-norm for some Gaussian random fields

L. V. Rozovskii

Abstract: In this paper we study the asymptotic behavior of the tail probability $P(V^2<r)$ as $r\to 0$, where the sum $V^2$ is given by the formula $V^2=a^2 \sum_{i,j\ge 1} (i+\beta)^{-2c}(j+\delta)^{-2}\xi^2_{ij}.$ Here $\{\xi_{ij}\}$ are independent standard Gaussian random variables, and $a>0$, $\beta >-1$, $\delta>-1$, $c>1/2$, $\ne 1$ are some constants. Thus, we study small deviations of the $L_2$-norm of certain two-parameter Gaussian random fields, that have the structure of a tensor product.

Keywords: small deviations, Karhunen–Loéve expansion, Gaussian random field, tensor product, $L_2$-norm.

 Funding Agency Grant Number Russian Foundation for Basic Research 16-01-00367

DOI: https://doi.org/10.4213/tvp5140

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English version:
Theory of Probability and its Applications, 2019, 63:3, 381–392

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Revised: 08.11.2017
Accepted:22.11.2017

Citation: L. V. Rozovskii, “On the exact asymptotics of small deviations of $L_2$-norm for some Gaussian random fields”, Teor. Veroyatnost. i Primenen., 63:3 (2018), 468–481; Theory Probab. Appl., 63:3 (2019), 381–392

Citation in format AMSBIB
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