Sharp conditions for the CLT of linear processes in a Hilbert Space

In this paper we study the behavior of sums of a linear process Xk = XJt _,o aj(Zk~j) associated to a strictly stationary sequence {c,k} kfl with values in a real separable Hilbert space and {ak}ksZ are linear operators from H to H. One of the results is that £"_, XJ^/n satisfies the CLT provided {i<} i s Z are i.i.d. centered having finite second moments and Y.J,-*. \\aj II LI in < & • We shall provide an example which shows that the condition on the operators is essen-tially sharp. Extensions of this result are given for sequences of weak dependent random variables { f t} k i Z, under minimal conditions. KEY WORDS: Central limit theorem; linear process in Hilbert space. 1