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Inconsistent marginal problem on finite sets. (English) Zbl 0907.60003
Beneš, Viktor (ed.) et al., Distributions with given marginals and moment problems. Proceedings of the 1996 conference, Prague, Czech Republic. Dordrecht: Kluwer Academic Publishers. 235-242 (1997).
Summary: An algorithm (Tetris) for finding a joint distribution $$P$$ (on a finite set) whose marginals are identical to given less-dimensional distributions (oligodistributions) was presented by the author [in: IPMU ‘96. Proc. 6th Int. IPMU Conf., Vol. II, 763-768 (1996)]. The problem, in a more general setting, is known as marginal problem [see H. G. Kellerer, Z. Wahrscheinlichkeitstheorie Verw. Geb. 3, 247-270 (1964; Zbl 0126.34003)]. The Tetris algorithm ends in a finite number of steps $$(O(nmK))$$ if the set of less-dimensional distributions is strongly consistent, i.e. at least one such $$P$$ exists. If it is not the case, i.e. the given distributions are not consistent, the fact is found in a finite number of steps, too. A new algorithm (Pentis), suggested in this paper, can be applied in this situation solving thus inconsistent marginal problem. It generates a distribution whose marginals differ a little (in absolute values) from the given oligodistributions. Being exponential in number of variables, both algorithms are effective for small-size marginal problems only.
For the entire collection see [Zbl 0885.00054].

##### MSC:
 60A10 Probabilistic measure theory
Zbl 0126.34003