ON THE EXACT ASYMPTOTICS OF SMALL DEVIATIONS OF L2-NORM FOR SOME GAUSSIAN RANDOM FIELDS

摘要

In this paper we study the asymptotic behavior of the tail probability P(V 2 < r) as r → 0, where the sum V~2 is given by the formula V~2 = a~2∑_(i,j≥1)(i+β)~(-2c)(j+δ)~(-2)ξ_(ij)~2. Here {ξ_(ij)} are independent standard Gaussian random variables, and a > 0, β > -1, δ > -1, c > 1/2, ≠ 1 are some constants. Thus, we study small deviations of the L2-norm of certain two-parameter Gaussian random fields, that have the structure of a tensor product.