## The Annals of Probability

### Existence of Probability Measures with Given Marginals

#### 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

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