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 ( μ ) given by X k = ∑ j = 0 ∞ ( j + 1 ) − D ε k − j for each k ≥ 1 , where D is defined by D f = { d ( s ) f ( s ) : s ∈ S } for each f ∈ L 2 ( μ ) with d : S → R and { ε k : k ∈ Z } are independent and identically distributed L 2 ( μ ) -valued random elements with E ε 0 = 0 and E ‖ ε 0 ‖ 2 < ∞ . We establish sufficient conditions for the functional central limit theorem for { X k : k ≥ 1 } when the series of operator norms ∑ j = 0 ∞ ‖ ( j + 1 ) − D ‖ diverges and show that the limit process generates an operator self-similar process.