Open Access
June, 1981 On the Law of Large Numbers
D. L. Hanson, Ralph P. Russo
Ann. Probab. 9(3): 513-519 (June, 1981). DOI: 10.1214/aop/1176994425


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.


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D. L. Hanson. Ralph P. Russo. "On the Law of Large Numbers." Ann. Probab. 9 (3) 513 - 519, June, 1981.


Published: June, 1981
First available in Project Euclid: 19 April 2007

zbMATH: 0466.60033
MathSciNet: MR614637
Digital Object Identifier: 10.1214/aop/1176994425

Primary: 60F15
Secondary: 60F10

Keywords: Law of Large Numbers , Strong law of large numbers

Rights: Copyright © 1981 Institute of Mathematical Statistics

Vol.9 • No. 3 • June, 1981
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