# 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

### Fingerprint

Moment Inequalities
Mixture of Distributions
Normality
Product-moment correlation
Passage Time
Scale Mixture
Birth-death Process
Kurtosis
Limit Theorems
Central limit theorem
Metric space
Arbitrary
Moment inequalities
Mixture of distributions
Limit theorems
Coefficients

### 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

### Cite this

Keilson, Julian ; Sleutel, F.W. / Mixtures of Distributions, Moment Inequalities and Measures of Exponentiality and Normality. In: Annals of probability. 1974 ; Vol. 2, No. 1. pp. 112-130.
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Mixtures of Distributions, Moment Inequalities and Measures of Exponentiality and Normality. / Keilson, Julian; Sleutel, F.W.

In: Annals of probability, Vol. 2, No. 1, 1974, p. 112-130.

TY - JOUR

T1 - Mixtures of Distributions, Moment Inequalities and Measures of Exponentiality and Normality

AU - Keilson, Julian

AU - Sleutel, F.W.

PY - 1974

Y1 - 1974

N2 - 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.

AB - 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.

KW - Weak convergence

KW - log-concavity

KW - log-convexity

KW - completely monotone densities

KW - total positivity

KW - measures of exponentiality and normality

KW - sojourn times

KW - moment inequalities

KW - Birth-death processes

KW - Mixtures of distributions

KW - IR-70392

U2 - 10.1214/aop/1176996756

DO - 10.1214/aop/1176996756

M3 - Article

VL - 2

SP - 112

EP - 130

JO - Annals of probability

JF - Annals of probability

SN - 0091-1798

IS - 1

ER -