On the Lower Tail of Gaussian Measures on l _p

Abstract

Throughout this paper $(\xi_k)_{k=1}^{\infty}$ denotes a sequence of independent, centered, Gaussian random variables with variance one. We shall study the behavior of

$$ P((\sum_{k\geq 1}a_{k}|\xi _{k}|^{p})^{1/p}\leq \varepsilon) $$

as ε → 0⁺ for p > 0 where $(a_{k})_{k=1}^{\infty}$ is a given sequence of positive numbers and $\sum_{k\geq1}a_{k}< +\infty$.