## The Annals of Probability

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

### Existence of Probability Measures with Given Marginals

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

#### 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.

#### Article information

**Source**

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

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

http://projecteuclid.org/euclid.aop/1176990236

**Digital Object Identifier**

doi:10.1214/aop/1176990236

**Mathematical Reviews number (MathSciNet)**

MR1127728

**Zentralblatt MATH identifier**

0739.60001

**JSTOR**

links.jstor.org

**Subjects**

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}

**Keywords**

Baseball tomography marginals

#### Citation

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. http://projecteuclid.org/euclid.aop/1176990236.