Suppose $X_n$ is an i.i.d. sequence of random variables with mean $\mu$ and that $t_n$ is a nondecreasing sequence of positive integers such that $t_n \leq n$. Let $S_n = X_1 + \cdots + X_n$. We give conditions under which $\max_{t_n \leq k \leq n} \big|\frac{S_n - S_{n - k}}{k} - \mu \big| \rightarrow 0$ almost surely and we discuss sharpness.
Publié le : 1981-06-14
Classification:
Law of large numbers,
strong law of large numbers,
60F15,
60F10
@article{1176994425,
author = {Hanson, D. L. and Russo, Ralph P.},
title = {On the Law of Large Numbers},
journal = {Ann. Probab.},
volume = {9},
number = {6},
year = {1981},
pages = { 513-519},
language = {en},
url = {http://dml.mathdoc.fr/item/1176994425}
}
Hanson, D. L.; Russo, Ralph P. On the Law of Large Numbers. Ann. Probab., Tome 9 (1981) no. 6, pp. 513-519. http://gdmltest.u-ga.fr/item/1176994425/