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Journal Article

# An Extended Version of the Erdos-Renyi Strong Law of Large Numbers

David M. Mason
The Annals of Probability
Vol. 17, No. 1 (Jan., 1989), pp. 257-265
Stable URL: http://www.jstor.org/stable/2244209
Page Count: 9

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## Abstract

Consider a sequence X1, X2, ..., of i.i.d. random variables. For each integer m ≥ 1 let Sm denote the mth partial sum of these random variables and set S0 = 0. Assuming that EX1 ≥ 0 and the moment generating function φ of X1 exists in a right neighborhood of 0 the Erdos-Renyi strong law of large numbers states that whenever k(n) is a sequence of positive integers such that log n/k(n) ∼ c as n → ∞, where $0 < c < \infty$ then max{(Sm + k(n) - Sm)/(γ(c)k(n)): 0 ≤ m ≤ n - k(n)} converges almost surely to 1, where γ(c) is a constant depending on c and φ. An extended version of this strong law is presented which shows that it remains true in a slightly altered form when log n/k(n) → ∞.

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