1 results
Coherent distributions on the square–extreme points and asymptotics
- Part of
-
- Journal:
- Journal of Applied Probability / Volume 61 / Issue 4 / December 2024
- Published online by Cambridge University Press:
- 05 April 2024, pp. 1240-1262
- Print publication:
- December 2024
-
- Article
-
- You have access
- HTML
- Export citation
-
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 
$\lim_{\alpha \to \infty}\alpha\cdot(\sup_{(X,Y)\in \mathcal{C}}\mathbb{E}|X-Y|^{\alpha}) = {2}/{\mathrm{e}}$.
