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The Annals of Probability
Let E be an infinite-dimensional vector space carrying a Gaussian measure μ with mean 0 and a measurable norm q. Let F(t):=μ(q⩽t). By a result of Borell, F is logarithmically concave. But we show that F′ may have infinitely many local maxima for norms q=supn|fn|/an where fn are independent standard normal variables. We also consider Hilbertian norms q=(Σbnf2n)12 with bn>0,Σbn<∞. Then as t↓0 we can have F(t)↓0 as rapidly as desired, or as slowly as any function which is o(tn) for all n. For bn=1/n2 and in a few closely related cases, we find the exact asymptotic behavior of F at 0. For more general bn we find inequalities bounding F between limits which are not too far apart.
Gaussian processes, seminorms, measure of small balls, lower tail distribution
Hoffmann-Jorgensen, J., Shepp, L. A., & Dudley, R. M. (1979). On the Lower Tail of Gaussian Seminorms. The Annals of Probability, 7 (2), 319-342. http://dx.doi.org/10.1214/aop/1176995091
Date Posted: 27 November 2017
This document has been peer reviewed.