由三级数定理导出的一类强极限定理

A Class of Strong Limit Theorems Induced Three Series Theorem

设 {an,n≥ 1 }是一正数列 ,{Xn,n≥ 1 }是一独立随机变量序列 ,{gn,n≥ 1 }是定义在 (-∞ ,+∞ )上的一列非降的正值偶函数 ,对于每个gn,存在pn >0 ,当｜x｜增加时有gn(x)｜x｜pn ↓ .若∑∞n =1Egn(Xn)gn(an)1qn <+∞ ,其中qn ≥ 1 ;0 <pn ≤ 1pn;pn >1 ,则∑∞n =1Xnana .s.

Let {a n,n≥1} be a sequence of positive numbe rs ,{X n,n≥1} be a sequence of independent random variables, {g n,n ≥1} be a sequence of positive even-functions non-de-creasing on (0,+∞) , for each g n,there exists p n>0,|x| increases, where as g n(x)|x| p n decreases. If ∑∞ ]n=1n=1Eg n(X n)g n(a n) 1q n <+∞,where q n≥1;0<p n≤1 p n;p n>1 ,Then ∑∞n=1x na n converge almost surely.