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June, 1973 A Note on the Rate of Convergence and Its Applications
Rasul A. Khan
Ann. Probab. 1(3): 504-508 (June, 1973). DOI: 10.1214/aop/1176996945


Let $S_n$ denote the partial sums of i.i.d. random variables with mean zero and moment generating function existing in some neighborhood of the origin. We give explicit upper bounds for $P_m^+ = P(S_n \geqq a + bn$ for some $n \geqq m)$ and $P_m = P(|S_n| \geqq a + bn$ for some $n \geqq m), a \geqq 0, b > 0$. These bounds immediately give the rate of convergence for the strong law of large numbers. An application is also made to a sequential selection procedure.


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Rasul A. Khan. "A Note on the Rate of Convergence and Its Applications." Ann. Probab. 1 (3) 504 - 508, June, 1973.


Published: June, 1973
First available in Project Euclid: 19 April 2007

zbMATH: 0263.60019
MathSciNet: MR350847
Digital Object Identifier: 10.1214/aop/1176996945

Primary: 60G50
Secondary: 60F99

Keywords: moment generating function , rate of convergence , sequential selection procedure , Strong law of large numbers , upper bound

Rights: Copyright © 1973 Institute of Mathematical Statistics

Vol.1 • No. 3 • June, 1973
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