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April, 1980 Precision Bounds for the Relative Error in the Approximation of $E|S_n|$ and Extensions
Michael J. Klass
Ann. Probab. 8(2): 350-367 (April, 1980). DOI: 10.1214/aop/1176994782


Let $S_n$ denote the $n$th partial sum of i.i.d. nonconstant mean zero random variables. Given an approximation $K(n)$ of $E|S_n|$, tight bounds are obtained for the ratio $E|S_n|/K(n)$. These bounds are best possible as $n$ tends to infinity. Implications of this result relate to the law of the iterated logarithm for mean zero variables, Chebyshev's inequality and Markov's inequality. Asymptotically exact lower-bounds are obtained for expectations of functions of row-sums of triangular arrays of independent but not necessarily identically distributed random variables. Expectations of "Poissonized random sums" are also treated.


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Michael J. Klass. "Precision Bounds for the Relative Error in the Approximation of $E|S_n|$ and Extensions." Ann. Probab. 8 (2) 350 - 367, April, 1980.


Published: April, 1980
First available in Project Euclid: 19 April 2007

zbMATH: 0428.60058
MathSciNet: MR566599
Digital Object Identifier: 10.1214/aop/1176994782

Primary: 60G50
Secondary: 26A86 , 60E05

Keywords: approximation of $n$-dimensional integrals , Chebyshev's inequality , expectation , integral representation , Law of the iterated logarithm , Markov's inequality

Rights: Copyright © 1980 Institute of Mathematical Statistics

Vol.8 • No. 2 • April, 1980
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