CLT FOR OPERATOR ABEL SUMS OF RANDOM ELEMENTS

Let (»k)k‚1 be a sequence of independent, identically distributed second order mean zero random elements in a separable Hilbert space H and A be an element of a certain class of linear continuous operators H ! H such that kAk < 1. Denote ·A := P1=0 A k »k. We prove that if kI i Ak tends to zero, where I is the identity operator, then the normalized sum (IiA 2 ) 1=2 ·A converges in distribution to a Gaussian random element.