The Annals of Probability

Existence of Probability Measures with Given Marginals

Sam Gutmann, J. H. B. Kemperman, J. A. Reeds, and L. A. Shepp

Full-text: Open access


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.

Article information

Ann. Probab. Volume 19, Number 4 (1991), 1781-1797.

First available in Project Euclid: 19 April 2007

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 52A40: Inequalities and extremum problems
Secondary: 28A35: Measures and integrals in product spaces 60A10: Probabilistic measure theory {For ergodic theory, see 28Dxx and 60Fxx}

Baseball tomography marginals


Gutmann, Sam; Kemperman, J. H. B.; Reeds, J. A.; Shepp, L. A. Existence of Probability Measures with Given Marginals. Ann. Probab. 19 (1991), no. 4, 1781--1797. doi:10.1214/aop/1176990236.

Export citation