Comparison results for the lower tail of Gaussian seminorms

Abstract

Let ξ=(ξ n ) be i.i.d.N(0, 1) random variables andq(x), q′(x):R ∞→[0, ∞) be seminorms. We investigate necessary and sufficient conditions that the ratio ofP(q(ξ)<e) andP(q′(ξ)<e) goes to a positive constant as e→0+. We give satisfactory answers forl 2-norms and also some results for sup-norms andl p-norms. Some applications are given to the rate of escape of infinite dimensional Brownian motion, and we give the lower tail of the Ornstein-Uhlenbeck process and a weighted Brownian bridge under theL 2-norms.