Operator self-similar processes and functional central limit theorems

Let {X-k : k >= 1) be a linear process with values in the separable Hilbert space L-2(mu) given by X-k = Sigma(infinity)(j=0)(j + 1) D epsilon k-j for each k >= 1, where D is defined by Df = {d (s) f (s) : s is an element of S) for each f is an element of L-2(mu) with d : S -> R and {epsilon(k) : k is an element of Z) are independent and identically distributed L-2(mu)-valued random elements with E epsilon(0) = 0 and E parallel to epsilon(0)parallel to(2) < infinity. We establish sufficient conditions for the functional central limit theorem for {X-k : k >= 1} when the series of operator norms Sigma(infinity)(j=0) parallel to(j + 1)parallel to(-D)parallel to diverges and show that the limit process generates an operator self-similar process. (C) 2014 Elsevier B.V. All rights reserved.