Complete Moment Convergence for Sung’s Type Weighted Sums of B-Valued Random Elements

Letp≥1/αand1/2<α≤1.Let{X,Xn, n≥1}be a sequence of independent and identically distributedB-valued random elements and let{ani, 1≤i≤n, n≥1}be an array of real numbers satisfying∑i=1naniq=O(n)for someq>p.We give necessary and sufficient conditions for complete moment convergence of the form∑n=1∞n(p-v)α-2E∑i=1naniXi-εnα+v<∞, ∀ε>0, where0