## The Annals of Probability

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

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

Julian Keilson and F. W. Steutel

#### 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

**Permanent link to this document**

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**

links.jstor.org

**Subjects**

Primary: 60E05: Distributions: general theory

Secondary: 60F05: Central limit and other weak theorems 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)

**Keywords**

Mixtures of distributions moment inequalities measures of exponentiality and normality birth-death processes weak convergence log-concavity log-convexity completely monotone densities total positivity sojourn times

#### Citation

Keilson, Julian; Steutel, F. W. Mixtures of Distributions, Moment Inequalities and Measures of Exponentiality and Normality. Ann. Probab. 2 (1974), no. 1, 112--130. doi:10.1214/aop/1176996756. https://projecteuclid.org/euclid.aop/1176996756