B值随机元阵列加权和的收敛性与大数定律

B值随机元阵列加权和的收敛性与大数定律 CONVERGENCE AND LAWS OF LARGE NUMBERS FOR WEIGHTED SUMS OF ARRAYS OF BANACH VALUED RANDOM ELEMENTS 甘师信 Gan Shixin 1 first-author 武汉大学数学系 武汉大学数学系 Department of Mathematics,Wuhan University Department of Mathematics,Wuhan University 令｛Ｘｎｉ，１≤ｉ≤ｋｎ↑∞，ｎ≥１｝为Ｂ值随机元阵列，｛ａｎｉ，１≤ｉ≤ｋｎ，ｎ≥１｝为实数阵列．讨论加权和Ｓｎ＝ｋｎｉ＝１ａｎｉＸｎｉ，ｎ≥１的收敛性．在条件ｓｕｐｎ，ｉＰ（Ｘｎｉ＞ｘ）＝０（ｘ－ｒ）下给出了一些收敛性结果（１≤ｒ＜ｐ≤２），同时用这种收敛性刻划了Ｂａｎａｃｈ空间的ｐ型性质 Convergence and laws of large numbers for weighted sums of arrays of Banach valued random elements are discused in this paper. The following results are proved. Let B be a seperable Banach space of tyep p,1<r<p≤2.{X ni ,1≤i≤k n↑∞,n≥1} is an array of rowwise independent Banach valued random elements with mean 0.{a ni ,1≤i≤k n↑∞,n≥1} is an array of real uumbers.If the following conditions are satisfied: i)\ sup n,i P（‖X ni ‖>x）=o（x -r ）; ii)\ lim n→∞ sup 1≤i≤k n | a ni |=0 and k ni=1|a ni | r≤C,n≥1,thenk ni=1a ni X ni P0If i),ii) and the follawing iii) hold. iii)\ sup n,i E‖X ni ‖ r<∞ ,then k ni=1a ni X ni L r 0,r∈（1,r）. \ \ B值随机元 p型空间 Lr收敛 依概率收敛 大数定律 Banach space valued random element p type space L r couvergence convergence in probability law of large numbers O211.4 国家自然科学基金,国家教委博士学科点基金 1997-05-01 2021-04-01 5