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

Roos, Bero

doi:10.1090/S0002-9939-2014-12429-4

On Bobkov’s approximate de Finetti representation via approximation of permanents of complex rectangular matrices
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Proc. Amer. Math. Soc. 143 (2015), 1785-1796 Request permission

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

S. Banach, Über homogene Polynome in $(L^2)$, Studia Math. 7 (1938), 36–44.
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, DOI 10.1214/009117904000000720
Sergey G. Bobkov, Generalized symmetric polynomials and an approximate de Finetti representation, J. Theoret. Probab. 18 (2005), no. 2, 399–412. MR 2137450, DOI 10.1007/s10959-005-3509-6
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, DOI 10.4310/MAA.2006.v13.n1.a1
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, DOI 10.1016/j.laa.2005.02.030
P. Diaconis and D. Freedman, Finite exchangeable sequences, Ann. Probab. 8 (1980), no. 4, 745–764. MR 577313
Seán Dineen, Complex analysis on infinite-dimensional spaces, Springer Monographs in Mathematics, Springer-Verlag London, Ltd., London, 1999. MR 1705327, DOI 10.1007/978-1-4471-0869-6
G. H. Hardy, J. E. Littlewood, and G. Pólya, Inequalities, Cambridge, at the University Press, 1952. 2d ed. MR 0046395
Lawrence A. Harris, Bernsteĭn’s polynomial inequalities and functional analysis, Irish Math. Soc. Bull. 36 (1996), 19–33. MR 1387033
Edwin Hewitt and Leonard J. Savage, Symmetric measures on Cartesian products, Trans. Amer. Math. Soc. 80 (1955), 470–501. MR 76206, DOI 10.1090/S0002-9947-1955-0076206-8
Lars Hörmander, On a theorem of Grace, Math. Scand. 2 (1954), 55–64. MR 62844, DOI 10.7146/math.scand.a-10395
O. D. Kellogg, On bounded polynomials in several variables, Math. Z. 27 (1928), no. 1, 55–64. MR 1544896, DOI 10.1007/BF01171085
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
D. S. Mitrinović, Analytic inequalities, Die Grundlehren der mathematischen Wissenschaften, Band 165, Springer-Verlag, New York-Berlin, 1970. In cooperation with P. M. Vasić. MR 0274686
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, DOI 10.1137/S0040585X9797821X
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, DOI 10.1137/S0040585X97978750