A central limit theorem for the linear process generated by associated random variables in a Hilbert space

Let {ξk,k∈Z}{ξk,k∈Z} be a strictly stationary associated sequence of HH-valued random variables with Eξk=0Eξk=0 and E‖ξk‖2<∞E‖ξk‖2<∞ and {ak,k∈Z}{ak,k∈Z} a sequence of linear operators such that ∑j=0∞‖aj‖L(H)<∞. For a linear process Xk=∑j=0∞ajξk−j we derive that ∑i=1nXi/n satisfies the central limit theorem.