## The Annals of Mathematical Statistics

### The Stochastic Independence of Symmetric and Homogeneous Linear and Quadratic Statistics

Eugene Lukacs

#### Abstract

The following theorem is proved. If a univariate distribution has moments of first and second order and admits a homogeneous and symmetric quadratic statistic $Q$ which is independently distributed of the mean of a sample of $n$ drawn from this distribution, then it is either the normal distribution ($Q$ is then proportional to the variance) or the degenerate distribution (in this case no restriction is imposed on $Q$) or a step function with two symmetrically located steps (in this case $Q$ is the sum of the squared observations). The converse of this statement is also true.

#### Article information

Source
Ann. Math. Statist. Volume 23, Number 3 (1952), 442-449.

Dates
First available in Project Euclid: 28 April 2007

https://projecteuclid.org/euclid.aoms/1177729389

Digital Object Identifier
doi:10.1214/aoms/1177729389

Mathematical Reviews number (MathSciNet)
MR50228

Zentralblatt MATH identifier
0047.38102

JSTOR