## The Annals of Probability

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

### On the Lower Tail of Gaussian Seminorms

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

#### Abstract

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

**Source**

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

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176995091

**Digital Object Identifier**

doi:10.1214/aop/1176995091

**Mathematical Reviews number (MathSciNet)**

MR525057

**Zentralblatt MATH identifier**

0424.60041

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60G15: Gaussian processes

Secondary: 60B99: None of the above, but in this section

**Keywords**

Gaussian processes seminorms measure of small balls lower tail distribution

#### Citation

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. https://projecteuclid.org/euclid.aop/1176995091