## The Annals of Probability

### Symmetrization and concentration inequalities for multilinear forms with applications to zero-one laws for Lévy chaos

#### Abstract

We consider stochastic processes $X=\{X(t),t\in T\}$ represented as a Lévy chaos of finite order, that is, as a finite sum of multiple stochastic integrals with respect to a symmetric infinitely divisible random measure. For a measurable subspace V of $R\sp T$ we prove a very general zero-one law $P(X\in V)=0$ or 1, providing a complete analogue to the corresponding situation in the case of symmetric infinitely divisible processes (single integrals with respect to an infinitely divisible random measure). Our argument requires developing a new symmetrization technique for multi-linear Rademacher forms, as well as generalizing Kanter's concentration inequality to multiple sums.

#### Article information

Source
Ann. Probab., Volume 24, Number 1 (1996), 422-437.

Dates
First available in Project Euclid: 15 January 2003

https://projecteuclid.org/euclid.aop/1042644724

Digital Object Identifier
doi:10.1214/aop/1042644724

Mathematical Reviews number (MathSciNet)
MR1387643

Zentralblatt MATH identifier
0854.60017

#### Citation

Rosiński, Jan; Samorodnitsky, Gennady. Symmetrization and concentration inequalities for multilinear forms with applications to zero-one laws for Lévy chaos. Ann. Probab. 24 (1996), no. 1, 422--437. doi:10.1214/aop/1042644724. https://projecteuclid.org/euclid.aop/1042644724