The Annals of Probability

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

Julian Keilson and F. W. Steutel

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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


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