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.
Publié le : 1980-04-14
Classification:
Expectation,
approximation of $n$-dimensional integrals,
integral representation,
law of the iterated logarithm,
Chebyshev's inequality,
Markov's inequality,
60G50,
60E05,
26A86
@article{1176994782,
author = {Klass, Michael J.},
title = {Precision Bounds for the Relative Error in the Approximation of $E|S\_n|$ and Extensions},
journal = {Ann. Probab.},
volume = {8},
number = {6},
year = {1980},
pages = { 350-367},
language = {en},
url = {http://dml.mathdoc.fr/item/1176994782}
}
Klass, Michael J. Precision Bounds for the Relative Error in the Approximation of $E|S_n|$ and Extensions. Ann. Probab., Tome 8 (1980) no. 6, pp. 350-367. http://gdmltest.u-ga.fr/item/1176994782/