Operator self-similar processes and functional central limit theorems

Let {Xk:k≥1}{Xk:k≥1} be a linear process with values in the separable Hilbert space L2(μ)L2(μ) given by Xk=∑j=0∞(j+1)−Dεk−j for each k≥1k≥1, where DD is defined by Df={d(s)f(s):s∈S}Df={d(s)f(s):s∈S} for each f∈L2(μ)f∈L2(μ) with d:S→Rd:S→R and {εk:k∈Z}{εk:k∈Z} are independent and identically distributed L2(μ)L2(μ)-valued random elements with Eε0=0Eε0=0 and E‖ε0‖2<∞E‖ε0‖2<∞. We establish sufficient conditions for the functional central limit theorem for {Xk:k≥1}{Xk: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.