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

Authors:	Florence Merlevède Magda Peligrad Sergey Utev

Abstract:	In 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.

Keywords:	Central limit theorem linear process in Hilbert space
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