By the method of Rio [10], for a locally square integrable periodic function f, we prove for almost every x and t > 0.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-cm90-2-9,
author = {Katusi Fukuyama},
title = {A version of the law of large numbers},
journal = {Colloquium Mathematicae},
volume = {89},
year = {2001},
pages = {295-298},
zbl = {0987.60040},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm90-2-9}
}
Katusi Fukuyama. A version of the law of large numbers. Colloquium Mathematicae, Tome 89 (2001) pp. 295-298. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm90-2-9/