论文信息 - On the n-Coupling Problem

On the n-Coupling Problem

In this paper we obtain based on an idea of M. Knott and C. S. Smith (1994, Linear Algebra Appl.199, 363?371) characterizations of solutions of three-coupling problems by reduction to the construction of optimal couplings of each of the variables to the sum. In the case of normal distributions this leads to a complete solution. Under a technical condition this idea also works for general distributions and one obtains explicit results. We extend these results to the n-coupling problem and derive a characterization of optimal n-couplings by several 2-coupling problems. This leads to some constructive existence results for Monge solutions.

L. Rüschendorf | Ludger Uckelmann

[1] L. Rüschendorf. Sharpness of Fréchet-bounds , 1981 .

[2] I. Olkin,et al. The distance between two random vectors with given dispersion matrices , 1982 .

[3] H. Kellerer. Duality theorems for marginal problems , 1984 .

[4] S. Rachev,et al. A characterization of random variables with minimum L 2 -distance , 1990 .

[5] M. Gelbrich. On a Formula for the L2 Wasserstein Metric between Measures on Euclidean and Hilbert Spaces , 1990 .

[6] Jim Freeman. Probability Metrics and the Stability of Stochastic Models , 1991 .

[7] L. Rüschendorf. FréChet-Bounds and Their Applications , 1991 .

[8] Svetlozar T. Rachev,et al. Maximum submatrix traces for positive definite matrices , 1993 .

[9] M. Knott,et al. On a generalization of cyclic monotonicity and distances among random vectors , 1994 .

[10] G. Alberti. On the structure of singular sets of convex functions , 1994 .

[11] L. Rüschendorf. Optimal solutions of multivariate coupling problems , 1995 .

[12] W. Gangbo,et al. The geometry of optimal transportation , 1996 .

[13] L. Rüschendorf,et al. On Optimal Multivariate Couplings , 1997 .

[14] Josef Štěpán,et al. Distributions with given marginals and moment problems , 1997 .

[15] W. Gangbo,et al. Optimal maps for the multidimensional Monge-Kantorovich problem , 1998 .