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.
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Li, W.V. Comparison results for the lower tail of Gaussian seminorms. J Theor Probab 5, 1–31 (1992). https://doi.org/10.1007/BF01046776
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DOI: https://doi.org/10.1007/BF01046776