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.
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Merlevède, F., Peligrad, M. & Utev, S. Sharp Conditions for the CLT of Linear Processes in a Hilbert Space. Journal of Theoretical Probability 10, 681–693 (1997). https://doi.org/10.1023/A:1022653728014
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DOI: https://doi.org/10.1023/A:1022653728014