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/