Probability and Mathematical Statistics: Vol. 11, Fasc. 1

COMPLEMENTS ON DECOUPLING INEQUALITIES FOR MULTILINEAR FUNCTIONS IN STABLE RANDOM VECTORS

Balram S. Rajput
Jan Rosinski

Abstract: Let $B$ and $V$ be real separable Banach spaces and $d$ be a positive integer. Let $M : Bd --> V$ be a measurable symmetric multilinear function, and let $X$ be a $B$ -valued symmetric $p$ -stable random vector. It is shown that if $0 < q < p/2,$ then the finiteness of $E ||M (X,...,X) ||q V$ is not sufficient for the validity of the important part of the decoupling inequalities. A natural condition, in terms of the spectral measure of $X$ and an algebraic equation involving $M,$ is proposed and it is proved that this condition ensures decoupling inequalities for all $q (- (0,p).$ This result complements de Acosta’s decoupling inequalities for multilinear functions in $B$ -valued symmetric $p$ -stable random vectors.

2000 AMS Mathematics Subject Classification: Primary: -; Secondary: -;

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