## The Annals of Mathematical Statistics

- Ann. Math. Statist.
- Volume 17, Number 1 (1946), 1-12.

### Some Distributions of Sample Means

George W. Brown and John W. Tukey

#### Abstract

It is shown that certain monomials in normally distributed quantities have stable distributions with index $2^{-k}$. This provides, for $k > 1$, simple examples where the mean of a sample has a distribution equivalent to that of a fixed, arbitrarily large multiple of a single observation. These examples include distributions symmetrical about zero, and positive distributions. Using these examples, it is shown that any distribution with a very long tail (of average order $\geq x^{-3/2}$) has the distributions of its sample means grow flatter and flatter as the sample size increases. Thus the sample mean provides less information than a single value. Stronger results are proved for still longer tails.

#### Article information

**Source**

Ann. Math. Statist., Volume 17, Number 1 (1946), 1-12.

**Dates**

First available in Project Euclid: 28 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aoms/1177731017

**Digital Object Identifier**

doi:10.1214/aoms/1177731017

**Mathematical Reviews number (MathSciNet)**

MR15745

**Zentralblatt MATH identifier**

0063.00626

**JSTOR**

links.jstor.org

#### Citation

Brown, George W.; Tukey, John W. Some Distributions of Sample Means. Ann. Math. Statist. 17 (1946), no. 1, 1--12. doi:10.1214/aoms/1177731017. https://projecteuclid.org/euclid.aoms/1177731017

#### See also

- Later Acknowledgment of Priority: Shashanka S. Mitra. Acknowledgment of Priority: "Distribution of Symmetric Stable Laws of Index $2^{-n}$" AND "Stable Laws of Index $2^{-n}$". Ann. Probab., Vol. 11, Iss. 2 (1983), 456.Project Euclid: euclid.aop/1176993612