On limit theorems for Banach-space-valued linear processes

Abstract

Let (X k) k∈Z be a linear process with values in a separable Hilbert space H given by X k = ∞ j=0 (j + 1) −N ε k−j for each k ∈ Z, where N : H → H is a bounded, linear normal operator and (ε k) k∈Z is a sequence of independent, identically distributed H-valued random variables with Eε 0 = 0 and Eε 0 2 < ∞. We investigate the central and the functional central limit theorem for (X k) k∈Z when the series of operator norms ∞ j=0 (j + 1) −N op diverges. Furthermore we show that the limit process in case of the functional central limit theorem generates an operator self-similar process.

The central limit theorem for a sequence of random processes with space-varying long memory*

“…then we can use Theorem 1 of Račkauskas and Suquet [8] to show that the central limit theorem holds for the partial sums of a similar second-order stationary sequence of L 2 (µ)-valued random elements. Suppose that {ψ k } is a sequence of independent and identically distributed random elements of L 2 (µ).…”

“…for each t ∈ S. The sequence ( 9) is essentially similar to the sequence {X k } of random processes (1). According to Theorem 1 of Račkauskas and Suquet [8], n −1/2 n k=1 Y k converges in distribution to a Gaussian random element of L 2 (µ) if n −1/2 n k=1 ψ k converges in distribution to a Gaussian random element of L 2 (µ) (see Račkauskas and Suquet [8] for more details).…”

On the CLT for discrete Fourier transforms of functional time series

“…For the regular partial sums process, this result compares to Račkauskas and Suquet (2010) who have studied partial sums of linear processes in Banach spaces. They show that the CLT for the innovations transfers to the linear process under summability of ( Ψ k L : k ≥ 0).…”

Operator self-similar processes and functional central limit theorems

“…Proof of Theorem 3. The convergence of Theorem 3 follows from Theorem 5 of Račkauskas and Suquet [18]…”