B值随机变量一类加权和的完全收敛性

A remark on complete convergence for weighted sums of B- valued random variables

设{Xni:1≤i≤n,n≥1}为行间独立的B值r.v.阵列,X为实值r.v.,E｜X｜p<∞,p>2,且对 x>0, 1≤i≤n,n≥1,都有P(‖Xni‖>x)≤P(｜X｜>x).{ani:1≤i≤n,n≥1}为满足条件∑ni=1a2ni=1,n≥1的实数阵列.则1n1 p∑ni=1aniXnip0蕴涵1n1 p∑ni=1aniXni完全收敛于0.

<Abstrcat>Let {Xni:1≤i≤n,n≥1} be an array of rowwise independent Bvalued random variables which is uniformly bounded by a random variable X satisfying E|X|p<∞ for some p>2.Let {ani:1≤i≤n,n≥1} be an array of real numbers satisfying ∑ni=1a2ni=1 for all n≥1.It is shown that if 1n1/p∑ni=1aniXni→0 in probability then 1n1/p∑ni=1aniXni converges completely to zero.