This paper establishes conditions that secure the almost sure upper and lower bounds for a particular normalized weighted sum of independent nonnegative random variables. These random variables do not possess a finite first moment so these results are not typical. These mild conditions allow us to show that the almost sure upper limit is infinity while the almost sure lower bound is one.
@article{bwmeta1.element.doi-10_1515_math-2017-0070,
author = {Andr\'e Adler},
title = {One sided strong laws for random variables with infinite mean},
journal = {Open Mathematics},
volume = {15},
year = {2017},
pages = {828-832},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_1515_math-2017-0070}
}
André Adler. One sided strong laws for random variables with infinite mean. Open Mathematics, Tome 15 (2017) pp. 828-832. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_1515_math-2017-0070/