随机级数sum from n=1 to ∞ X_n(ω)f_n(x)的收敛性

Convergence of Random Series sum from n=1 to ∞ X_n(ω)f_n(x)

运用简化原理，得到了对称随机级数∑n=1^∞Xn（ω）fn（x）若在Lω^2中a．s．收敛或Cesaro有界，则它关于dω^-（x）几乎必然几乎处处收敛的结果，并给出一反例，说明这个结果的逆是不正确的．然后研究了在一般的情况下，当随机系数{Xn}满足“A↓n〉0,EXn=0,aE1/2｜Xn｜^2≤E｜Xn｜〈∞”的条件下，该级数收敛的充分必要条件．

This paper firstly use reduction principle to obtain a good result ： if random series ∑n=1^∞Xn（ω）fn（x） is convergent or Cesaro-bounded in Lω^2,when random coefficients {Xn } are symmetric, it is almost surely everywhere convergent on dω^-（x）. We can see the inverse of the result is incorrect by giving an example. And then this paper obtains a necessary and sufficient condition on convergence of series without symmetric hypothesis.