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Coherent distributions on the square–extreme points and asymptotics
Part of:
Distribution theory - Probability
Published online by Cambridge University Press: 05 April 2024
Abstract
Let 
$\mathcal{C}$ denote the family of all coherent distributions on the unit square 
$[0,1]^2$, i.e. all those probability measures 
$\mu$ for which there exists a random vector 
$(X,Y)\sim \mu$, a pair 
$(\mathcal{G},\mathcal{H})$ of 
$\sigma$-fields, and an event E such that 
$X=\mathbb{P}(E\mid\mathcal{G})$, 
$Y=\mathbb{P}(E\mid\mathcal{H})$ almost surely. We examine the set 
$\mathrm{ext}(\mathcal{C})$ of extreme points of 
$\mathcal{C}$ and provide its general characterisation. Moreover, we establish several structural properties of finitely-supported elements of 
$\mathrm{ext}(\mathcal{C})$. We apply these results to obtain the asymptotic sharp bound 
$\lim_{\alpha \to \infty}\alpha\cdot(\sup_{(X,Y)\in \mathcal{C}}\mathbb{E}|X-Y|^{\alpha}) = {2}/{\mathrm{e}}$.
MSC classification
Primary:
60E05: Distributions
Secondary:
60E15: Inequalities; stochastic orderings
- Type
- Original Article
- Information
- Copyright
- © The Author(s), 2024. Published by Cambridge University Press on behalf of Applied Probability Trust
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