scholarly journals Convergence in the pth-mean and some Weak Laws of Large Numbers for weighted sums of random elements in separable normed linear spaces

1984 ◽  
Vol 15 (1) ◽  
pp. 124-134 ◽  
Author(s):  
Xiang Chen Wang ◽  
M Bhaskara Rao
Keyword(s):  
Weighted Sums ◽  
Linear Spaces ◽  
Laws Of Large Numbers ◽  
Normed Linear Spaces ◽  
Large Numbers ◽  
Random Elements ◽  
Weak Laws

scholarly journals On the almost sure convergence of weighted sums of random elements inD[0,1]

1981 ◽  
Vol 4 (4) ◽  
pp. 745-752
Author(s):  
R. L. Taylor ◽  
C. A. Calhoun
Keyword(s):  
Almost Sure Convergence ◽  
Growth Conditions ◽  
Weighted Sums ◽  
Integral Conditions ◽  
Laws Of Large Numbers ◽  
Strong Laws ◽  
Convergence Results ◽  
Large Numbers ◽  
Random Elements

Let{wn}be a sequence of positive constants andWn=w1+…+wnwhereWn→∞andwn/Wn→∞. Let{Wn}be a sequence of independent random elements inD[0,1]. The almost sure convergence ofWn−1∑k=1nwkXkis established under certain integral conditions and growth conditions on the weights{wn}. The results are shown to be substantially stronger than the weighted sums convergence results of Taylor and Daffer (1980) and the strong laws of large numbers of Ranga Rao (1963) and Daffer and Taylor (1979).

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