On complete moment convergence for CAANA random vectors in Hilbert spaces

For a sequence {Xn,n≥1} of coordinatewise asymptotically almost negatively associated random vectors in Hilbert space, we investigate the convergence of ∑n=1∞1n1+αE(max1≤k≤n‖Sk‖−ϵnα)+ and ∑n=1∞lognn1+αE(max1≤k≤n‖Sk‖−ϵnα)+, where Sk=∑i=1kXi and a+=max{a,0}. This investigation provides the complete moment convergence for the case αr=1.