Complete Moment Convergence for Sung’s Type Weighted Sums ofB-Valued Random Elements
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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<v<p.A strong law of large numbers for weighted sums of independentB-valued random elements is also obtained.
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