Sharp Conditions for the CLT of Linear Processes in a Hilbert Space

AbstractIn this paper we study the behavior of sums of a linear process $$X_k = \sum {_{j = - \infty }^\infty } a_j (\xi _{k - j} )$$ associated to a strictly stationary sequence $$\{ \xi _k \} _{k \in \mathbb{Z}} $$ with values in a real separable Hilbert space and $$\{ a_k \} _{k \in \mathbb{Z}} $$ are linear operators from H to H. One of the results is that $$\sum {_{i = 1}^n } X_i /\sqrt n $$ satisfies the CLT provided $$\{ \xi _k \} _{k \in \mathbb{Z}} $$ are i.i.d. centered having finite second moments and $$\sum {_{j = - \infty }^\infty } \left\| {a_j } \right\|_{L(H)} < \infty $$ . We shall provide an example which shows that the condition on the operators is essentially sharp. Extensions of this result are given for sequences of weak dependent random variables $$\{ \xi _k \} _{k \in \mathbb{Z}} $$ under minimal conditions.