On the central limit theorem in product spaces

Abstract

Suppose thatE andF are separable Banach spaces,X andY are independent symmetricE andF-valued random vectors respectively. This paper is devoted to the study of the central limit theorem forX ◯Y in the injective and projective tensor product spacesE $$\mathop \otimes \limits^ \vee $$ F andE $$\mathop \otimes \limits^ \wedge $$ F. Special attention is paid tol 2 $$\mathop \otimes \limits^ \wedge $$ l 2. In addition, two counter-examples are given.