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

Julian Keilson, F.W. Sleutel

## 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.
Original language English 112-130 Annals of probability 2 1 https://doi.org/10.1214/aop/1176996756 Published - 1974

## Keywords

• Weak convergence
• log-concavity
• log-convexity
• completely monotone densities
• total positivity
• measures of exponentiality and normality
• sojourn times
• moment inequalities
• Birth-death processes
• Mixtures of distributions
• IR-70392