## The Annals of Probability

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

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

#### 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

**Permanent link to this document**

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**

links.jstor.org

**Subjects**

Primary: 60G50: Sums of independent random variables; random walks

Secondary: 60E05: Distributions: general theory 26A86

**Keywords**

Expectation approximation of $n$-dimensional integrals integral representation law of the iterated logarithm Chebyshev's inequality Markov's inequality

#### Citation

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