On the almost sure convergence for a linear process generated by negatively associated random variables in a Hilbert space

Let {ξk,k∈Z} be a strictly stationary negatively associated sequence of H-valued random variables with Eξk=0,E‖ξk‖<∞ and E‖ξk‖2<∞ and {ak,k∈Z} a sequence of linear operators such that ∑j=0∞j‖aj‖L(H)<∞. For a linear process Xk=∑j=0∞ajξk−j we derive that n−1∑k=1nXk→0 almost surely as n→∞.