Open Access
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


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.


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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.


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

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

Primary: 52A40
Secondary: 28A35 , 60A10

Keywords: baseball , marginals , tomography

Rights: Copyright © 1991 Institute of Mathematical Statistics

Vol.19 • No. 4 • October, 1991
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