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April, 1979 On the Lower Tail of Gaussian Seminorms
J. Hoffmann-Jorgensen, L. A. Shepp, R. M. Dudley
Ann. Probab. 7(2): 319-342 (April, 1979). DOI: 10.1214/aop/1176995091


Let $E$ be an infinite-dimensional vector space carrying a Gaussian measure $\mu$ with mean 0 and a measurable norm $q$. Let $F(t) := \mu(q \leqslant t)$. By a result of Borell, $F$ is logarithmically concave. But we show that $F'$ may have infinitely many local maxima for norms $q = \sup_n|f_n|/a_n$ where $f_n$ are independent standard normal variables. We also consider Hilbertian norms $q = (\Sigma b_nf^2_n)^{\frac{1}{2}}$ with $b_n > 0, \Sigma b_n < \infty$. Then as $t \downarrow 0$ we can have $F(t) \downarrow 0$ as rapidly as desired, or as slowly as any function which is $o(t^n)$ for all $n$. For $b_n = 1/n^2$ and in a few closely related cases, we find the exact asymptotic behavior of $F$ at 0. For more general $b_n$ we find inequalities bounding $F$ between limits which are not too far apart.


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J. Hoffmann-Jorgensen. L. A. Shepp. R. M. Dudley. "On the Lower Tail of Gaussian Seminorms." Ann. Probab. 7 (2) 319 - 342, April, 1979.


Published: April, 1979
First available in Project Euclid: 19 April 2007

zbMATH: 0424.60041
MathSciNet: MR525057
Digital Object Identifier: 10.1214/aop/1176995091

Primary: 60G15
Secondary: 60B99

Keywords: Gaussian processes , lower tail distribution , measure of small balls , seminorms

Rights: Copyright © 1979 Institute of Mathematical Statistics


Vol.7 • No. 2 • April, 1979
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