Small ball probabilities for certain gaussian fields

TL;DR

This paper analyzes the asymptotic behavior of small ball probabilities for a specific class of Gaussian fields defined by a double sum involving independent normal variables, focusing on the tail probability as the sum approaches zero.

Contribution

It provides a detailed study of the tail behavior of a particular Gaussian field's small ball probabilities, extending understanding of such probabilities for complex Gaussian structures.

Findings

Derived asymptotic formulas for tail probabilities as r approaches 0.

Identified the influence of parameters b and δ on probability decay.

Enhanced theoretical understanding of Gaussian field small deviations.

Abstract

We study the behavior of the tail probabilities $P(V^2<r)$ as $r\to 0$, where $V^2$ is defined by the following double sum $$ V^2 =\pi^{-4}\,\sum\limits_{i,j\ge 1} \Big((i+b)\,(j + \delta)\Big)^{-2}\,\xi_{ij}^2,$$ where $ \{\xi_{ij}\} $ are independent standard normal random variables, and $b$ and $\delta$ are constants: $b>-1,\ \delta>-1$.