Date of this Version
The Annals of Probability
We show that if f is a probability density on Rn wrt Lebesgue measure (or any absolutely continuous measure) and 0 ≤ f ≤ 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 ≤ X, Y ≤ 1, we further study functions 0 ≤ ϕ ≤ 1 such that Z = ϕ (X,Y) satisfies E(Z|X) = X and E(Z|Y) = Y. If there is a solution then there also is a nondecreasing solution ϕ(x,y). These results are applied to tomography and baseball.
baseball, tomography, marginals
Gutmann, S., Kemperman, J. B., Reeds, J. A., & Shepp, L. A. (1991). Existence of Probability Measures With Given Marginals. The Annals of Probability, 19 (4), 1781-1797. http://dx.doi.org/10.1214/aop/1176990236
Date Posted: 27 November 2017
This document has been peer reviewed.