Let $\{X_n\}$ be random elements in a separable Banach space which is $p$-smoothable and let $\{a_k\}$ and $\{A_k\}$ denote positive random variables such that almost surely $A_k$ is monotonically increasing to $\infty$ and that $A_k/a_k \rightarrow \infty$. Convergence almost surely is obtained for the weighted sum $A^{-1}_n \sum^n_{k=1} a_kX_k$ and is related to a moment condition on the random elements and a growth condition on the random weights.
Publié le : 1983-08-14
Classification:
Weighted sums,
random weights,
laws of large numbers,
$p$-smoothable,
60B12,
60B11
@article{1176993524,
author = {Taylor, R. L. and Calhoun, C. A.},
title = {On the Almost Sure Convergence of Randomly Weighted Sums of Random Elements},
journal = {Ann. Probab.},
volume = {11},
number = {4},
year = {1983},
pages = { 795-797},
language = {en},
url = {http://dml.mathdoc.fr/item/1176993524}
}
Taylor, R. L.; Calhoun, C. A. On the Almost Sure Convergence of Randomly Weighted Sums of Random Elements. Ann. Probab., Tome 11 (1983) no. 4, pp. 795-797. http://gdmltest.u-ga.fr/item/1176993524/