Article

Journal of Theoretical Probability

, Volume 5, Issue 1, pp 1-31

First online:

Comparison results for the lower tail of Gaussian seminorms

  • Wenbo V. LiAffiliated withDepartment of Mathematics, University of Wisconsin-Madison

Rent the article at a discount

Rent now

* Final gross prices may vary according to local VAT.

Get Access

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)<ε) andP(q′(ξ)<ε) goes to a positive constant as ε→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.

Key Words

Lower tail Gaussian seminorms Gaussian processes infinite dimensional Brownian motion