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

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...

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Bibliographic Details
Published in:The Annals of probability 1974-02, Vol.2 (1), p.112-130
Main Authors: Keilson, Julian, Steutel, F. W.
Format: Article
Language:English
Subjects:
60E05
60F05
60J80
birth-death processes
Central limit theorem
completely monotone densities
Distance functions
Gaussian distributions
Kurtosis
log-concavity
log-convexity
Mathematical moments
Mathematical monotonicity
measures of exponentiality and normality
Mixtures of distributions
moment inequalities
Perceptron convergence procedure
Probability theory
Random variables
sojourn times
total positivity
weak convergence
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Summary: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.
ISSN:0091-1798
2168-894X
DOI:10.1214/aop/1176996756