Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Multinomial probabilities, permanents and a conjecture of Karlin and Rinott


Author: R. B. Bapat
Journal: Proc. Amer. Math. Soc. 102 (1988), 467-472
MSC: Primary 15A15; Secondary 60E15
MathSciNet review: 928962
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The probability density function of a multiparameter multinomial distribution can be expressed in terms of the permanent of a suitable matrix. This fact and certain results on conditionally negative definite matrices are used to prove a conjecture due to Karlin and Rinott.


References [Enhancements On Off] (What's this?)

  • [1] A. D. Alexandroff, Zur theorie der gemischten volumina von konvexen körpern. IV, Mat. Sb. 45 (1938), no. 3, 227-251. (Russian; German summary).
  • [2] G. P. Egorychev, The solution of van der Waerden’s problem for permanents, Adv. in Math. 42 (1981), no. 3, 299–305. MR 642395, 10.1016/0001-8708(81)90044-X
  • [3] D. I. Falikman, Proof of the van der Waerden conjecture on the permanent of a doubly stochastic matrix, Mat. Zametki 29 (1981), no. 6, 931–938, 957 (Russian). MR 625097
  • [4] Samuel Karlin and Yosef Rinott, Entropy inequalities for classes of probability distributions. II. The multivariate case, Adv. in Appl. Probab. 13 (1981), no. 2, 325–351. MR 612207, 10.2307/1426687
  • [5] M. Marcus and M. Newman, The permanent of a symmetric matrix, Notices Amer. Math. Soc. 8 (1981), 595.
  • [6] Charles A. Micchelli, Interpolation of scattered data: distance matrices and conditionally positive definite functions, Constr. Approx. 2 (1986), no. 1, 11–22. MR 891767, 10.1007/BF01893414
  • [7] Henryk Minc, Permanents, Encyclopedia of Mathematics and its Applications, vol. 6, Addison-Wesley Publishing Co., Reading, Mass., 1978. With a foreword by Marvin Marcus. MR 504978
  • [8] K. R. Parthasarathy and K. Schmidt, Positive definite kernels, continuous tensor products, and central limit theorems of probability theory, Lecture Notes in Mathematics, Vol. 272, Springer-Verlag, Berlin-New York, 1972. MR 0622034
  • [9] Richard Sinkhorn, A relationship between arbitrary positive matrices and doubly stochastic matrices, Ann. Math. Statist. 35 (1964), 876–879. MR 0161868
  • [10] J. H. van Lint, Notes on Egoritsjev’s proof of the van der Waerden conjecture, Linear Algebra Appl. 39 (1981), 1–8. MR 625232, 10.1016/0024-3795(81)90285-8
  • [11] D. V. Widder, The Laplace transform, Princeton Univ. Press, Princeton, N.J., 1946.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 15A15, 60E15

Retrieve articles in all journals with MSC: 15A15, 60E15


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1988-0928962-9
Keywords: Permanents, multinomial distribution, conditionally negative definite matrices
Article copyright: © Copyright 1988 American Mathematical Society