## The Annals of Probability

### Precision Bounds for the Relative Error in the Approximation of $E|S_n|$ and Extensions

Michael J. Klass

#### Abstract

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.

#### Article information

Source
Ann. Probab., Volume 8, Number 2 (1980), 350-367.

Dates
First available in Project Euclid: 19 April 2007

https://projecteuclid.org/euclid.aop/1176994782

Digital Object Identifier
doi:10.1214/aop/1176994782

Mathematical Reviews number (MathSciNet)
MR566599

Zentralblatt MATH identifier
0428.60058

JSTOR
Klass, Michael J. Precision Bounds for the Relative Error in the Approximation of $E|S_n|$ and Extensions. Ann. Probab. 8 (1980), no. 2, 350--367. doi:10.1214/aop/1176994782. https://projecteuclid.org/euclid.aop/1176994782