The Annals of Probability

On the Lower Tail of Gaussian Seminorms

J. Hoffmann-Jorgensen, L. A. Shepp, and R. M. Dudley

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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.

Article information

Ann. Probab. Volume 7, Number 2 (1979), 319-342.

First available in Project Euclid: 19 April 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 60G15: Gaussian processes
Secondary: 60B99: None of the above, but in this section

Gaussian processes seminorms measure of small balls lower tail distribution


Hoffmann-Jorgensen, J.; Shepp, L. A.; Dudley, R. M. On the Lower Tail of Gaussian Seminorms. Ann. Probab. 7 (1979), no. 2, 319--342. doi:10.1214/aop/1176995091.

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