Open Access
Translator Disclaimer
October, 1991 Existence of Probability Measures with Given Marginals
Sam Gutmann, J. H. B. Kemperman, J. A. Reeds, L. A. Shepp
Ann. Probab. 19(4): 1781-1797 (October, 1991). DOI: 10.1214/aop/1176990236

Abstract

We show that if $f$ is a probability density on $R^n$ wrt Lebesgue measure (or any absolutely continuous measure) and $0 \leq f \leq 1$, then there is another density $g$ with only the values 0 and 1 and with the same $(n - 1)$-dimensional marginals in any finite number of directions. This sharpens, unifies and extends the results of Lorentz and of Kellerer. Given a pair of independent random variables $0 \leq X,Y \leq 1$, we further study functions $0 \leq \phi \leq 1$ such that $Z = \phi(X,Y)$ satisfies $E(Z\mid X) = X$ and $E(Z\mid Y) = Y$. If there is a solution then there also is a nondecreasing solution $\phi(x,y)$. These results are applied to tomography and baseball.

Citation

Download Citation

Sam Gutmann. J. H. B. Kemperman. J. A. Reeds. L. A. Shepp. "Existence of Probability Measures with Given Marginals." Ann. Probab. 19 (4) 1781 - 1797, October, 1991. https://doi.org/10.1214/aop/1176990236

Information

Published: October, 1991
First available in Project Euclid: 19 April 2007

zbMATH: 0739.60001
MathSciNet: MR1127728
Digital Object Identifier: 10.1214/aop/1176990236

Subjects:
Primary: 52A40
Secondary: 28A35 , 60A10

Keywords: baseball , marginals , tomography

Rights: Copyright © 1991 Institute of Mathematical Statistics

JOURNAL ARTICLE
17 PAGES


SHARE
Vol.19 • No. 4 • October, 1991
Back to Top