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

Abstract EN

Let p ≥ 1 / α and 1 / 2 < α ≤ 1 .

Let { X , X n , n ≥ 1 } be a sequence of independent and identically distributed B -valued random elements and let { a n i , 1 ≤ i ≤ n , n ≥ 1 } be an array of real numbers satisfying ∑ i = 1 n a n i q = O ( n ) for some q > p .

We give necessary and sufficient conditions for complete moment convergence of the form ∑ n = 1 ∞ n ( p - v ) α - 2 E ∑ i = 1 n a n i X i - ε n α + v < ∞ , ∀ ε > 0 , where 0 < v < p .

A strong law of large numbers for weighted sums of independent B -valued random elements is also obtained.