Asymptotic normality of sums of Hilbert space valued random elements

Abstract

Abstract We investigate the asymptotic normality of distributions of the sequence ∑ k ∈ ℤ u n , k ⁢ X k {\sum_{k\in\mathbb{Z}}u_{n,k}X_{k}} , n ∈ ℕ {n\in\mathbb{N}} , where ( X k , k ∈ ℤ ) {(X_{k},k\in\mathbb{Z})} either is a sequence of i.i.d. random elements or constitutes a linear process with i.i.d. innovations in a separable Hilbert space. The weights ( u n , k ) {(u_{n,k})} are in general a family of linear bounded operators. This model includes operator weighted sums of Hilbert space valued linear processes, operator-wise discounted sums in a Hilbert space as well some extensions of classical summation methods.