Journal of Theoretical Probability

, Volume 5, Issue 1, pp 1–31

Comparison results for the lower tail of Gaussian seminorms

  • Wenbo V. Li
Article

DOI: 10.1007/BF01046776

Cite this article as:
Li, W.V. J Theor Probab (1992) 5: 1. doi:10.1007/BF01046776

Abstract

Let ξ=(ξn) 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 forl2-norms and also some results for sup-norms andlp-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 theL2-norms.

Key Words

Lower tailGaussian seminormsGaussian processesinfinite dimensional Brownian motion

Copyright information

© Plenum Publishing Corporation 1992

Authors and Affiliations

  • Wenbo V. Li
    • 1
  1. 1.Department of MathematicsUniversity of Wisconsin-MadisonMadison