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)

Request Permissions   Purchase Content 


On Bobkov's approximate de Finetti representation via approximation of permanents of complex rectangular matrices

Author: Bero Roos
Journal: Proc. Amer. Math. Soc. 143 (2015), 1785-1796
MSC (2010): Primary 60G09, 62E17, 15A45
Published electronically: December 4, 2014
MathSciNet review: 3314090
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Bobkov (J. Theoret. Probab. 18(2) (2005) 399-412) investigated an approximate de Finetti representation for probability measures, on product measurable spaces, which are symmetric under permutations of coordinates. One of the main results of that paper was an explicit approximation bound for permanents of complex rectangular matrices, which was shown by a somewhat complicated induction argument. In this paper, we indicate how to avoid the induction argument using an (asymptotic) expansion. Our approach makes it possible to give new explicit higher order approximation bounds for such permanents and in turn for the probability measures mentioned above.

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

  • [1] S. Banach, Über homogene Polynome in $ (L^2)$, Studia Math. 7 (1938), 36-44.
  • [2] Sergey G. Bobkov, Concentration of normalized sums and a central limit theorem for noncorrelated random variables, Ann. Probab. 32 (2004), no. 4, 2884-2907. MR 2094433 (2005i:60041),
  • [3] Sergey G. Bobkov, Generalized symmetric polynomials and an approximate de Finetti representation, J. Theoret. Probab. 18 (2005), no. 2, 399-412. MR 2137450 (2006c:28003),
  • [4] Eric Carlen, Elliott H. Lieb, and Michael Loss, An inequality of Hadamard type for permanents, Methods Appl. Anal. 13 (2006), no. 1, 1-17. MR 2275869 (2007k:15013),
  • [5] Gi-Sang Cheon and Ian M. Wanless, An update on Minc's survey of open problems involving permanents, Linear Algebra Appl. 403 (2005), 314-342. MR 2140290 (2006b:15012),
  • [6] P. Diaconis and D. Freedman, Finite exchangeable sequences, Ann. Probab. 8 (1980), no. 4, 745-764. MR 0577313 (81m:60032)
  • [7] Seán Dineen, Complex analysis on infinite-dimensional spaces, Springer Monographs in Mathematics, Springer-Verlag London Ltd., London, 1999. MR 1705327 (2001a:46043)
  • [8] G. H. Hardy, J. E. Littlewood, and G. Pólya, Inequalities, Cambridge, at the University Press, 1952. 2d ed. MR 0046395 (13,727e)
  • [9] Lawrence A. Harris, Bernsteĭn's polynomial inequalities and functional analysis, Irish Math. Soc. Bull. 36 (1996), 19-33. MR 1387033 (97h:46070)
  • [10] Edwin Hewitt and Leonard J. Savage, Symmetric measures on Cartesian products, Trans. Amer. Math. Soc. 80 (1955), 470-501. MR 0076206 (17,863g)
  • [11] Lars Hörmander, On a theorem of Grace, Math. Scand. 2 (1954), 55-64. MR 0062844 (16,27b)
  • [12] O. D. Kellogg, On bounded polynomials in several variables, Math. Z. 27 (1928), no. 1, 55-64. MR 1544896,
  • [13] Henryk Minc, Permanents, Encyclopedia of Mathematics and its Applications, vol. 9999, Addison-Wesley Publishing Co., Reading, Mass., 1978. With a foreword by Marvin Marcus. MR 0504978 (80d:15009)
  • [14] D. S. Mitrinović, Analytic inequalities, Springer-Verlag, New York, 1970. In cooperation with P. M. Vasić. Die Grundlehren der mathematischen Wissenschaften, Band 165. MR 0274686 (43 #448)
  • [15] B. Roos, Binomial approximation to the Poisson binomial distribution: the Krawtchouk expansion, Teor. Veroyatnost. i Primenen. 45 (2000), no. 2, 328-344 (English, with Russian summary); English transl., Theory Probab. Appl. 45 (2001), no. 2, 258-272. MR 1967760 (2003k:60099),
  • [16] B. Roos, Multinomial and Krawtchouk approximations to the generalized multinomial distribution, Teor. Veroyatnost. i Primenen. 46 (2001), no. 1, 117-133 (English, with Russian summary); English transl., Theory Probab. Appl. 46 (2002), no. 1, 103-117. MR 1968708 (2004b:62043),

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 60G09, 62E17, 15A45

Retrieve articles in all journals with MSC (2010): 60G09, 62E17, 15A45

Additional Information

Bero Roos
Affiliation: FB IV – Department of Mathematics, University of Trier, 54286 Trier, Germany

Keywords: de Finetti representation, permanent, Hadamard type inequality
Received by editor(s): March 13, 2012
Received by editor(s) in revised form: September 20, 2013
Published electronically: December 4, 2014
Communicated by: Walter Craig
Article copyright: © Copyright 2014 American Mathematical Society

American Mathematical Society