In this paper, we study weighted sums $\sum^n_{i=1} c_{n-i} X_i$ of i.i.d. zero-mean random variables $X_1, X_2, \cdots$, under the condition that the sequence $(c_n)$ is square summable. It is proved that such weighted sums are, with probability 1, of smaller order than $n^{1/\alpha}$ (respectively $\log n$, etc.) $\operatorname{iff} E|X_1|^\alpha < \infty$ (respectively $Ee^{t|X_1|} < \infty$ for all $t < \infty$, etc.). Certain analogs of the law of the iterated logarithm for such weighted sums are also obtained.
Publié le : 1973-10-14
Classification:
Weighted sums,
strong law of large numbers,
Marcinkiewicz-Zygmund inequalities,
symmetrization,
exponential bounds,
law of the iterated logarithm,
coin tossing,
Khintchine inequality,
success runs,
double arrays,
60J30,
60F15
@article{1176996847,
author = {Chow, Y. S. and Lai, T. L.},
title = {Limiting Behavior of Weighted Sums of Independent Random Variables},
journal = {Ann. Probab.},
volume = {1},
number = {5},
year = {1973},
pages = { 810-824},
language = {en},
url = {http://dml.mathdoc.fr/item/1176996847}
}
Chow, Y. S.; Lai, T. L. Limiting Behavior of Weighted Sums of Independent Random Variables. Ann. Probab., Tome 1 (1973) no. 5, pp. 810-824. http://gdmltest.u-ga.fr/item/1176996847/