## The Annals of Probability

### Mixtures of Distributions, Moment Inequalities and Measures of Exponentiality and Normality

#### Abstract

The central limit theorem and limit theorems for rarity require measures of normality and exponentiality for their implementation. Simple useful measures are exhibited for these in a metric space setting, obtained from inequalities for scale mixtures and power mixtures. It is shown that the Pearson coefficient of Kurtosis is such a measure for normality in a broad class $\mathscr{D}$ containing most of the classical distributions as well as the passage time densities $s_{mn}(\tau)$ for arbitrary birth-death processes.

#### Article information

Source
Ann. Probab., Volume 2, Number 1 (1974), 112-130.

Dates
First available in Project Euclid: 19 April 2007

https://projecteuclid.org/euclid.aop/1176996756

Digital Object Identifier
doi:10.1214/aop/1176996756

Mathematical Reviews number (MathSciNet)
MR356180

Zentralblatt MATH identifier
0325.60019

JSTOR