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

We consider stochastic processes X=(t),t∈ 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 T we prove a very general zero-one law P(X∈ 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.