Abstract
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.
Citation
D. L. Hanson. Ralph P. Russo. "On the Law of Large Numbers." Ann. Probab. 9 (3) 513 - 519, June, 1981. https://doi.org/10.1214/aop/1176994425
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