Riesz measures and Wishart laws associated to quadratic maps. (English) Zbl 1284.62314
Summary: We introduce a natural definition of Riesz measures and Wishart laws associated to an \(\Omega\)-positive (virtual) quadratic map, where \(\Omega \subset R^n\) is a regular open convex cone. In this context we prove new general formulas for moments of the Wishart laws on non-symmetric cones. For homogeneous cases, all the quadratic maps are characterized and the associated Riesz measure and Wishart law with its moments are described explicitly. We apply the theory of relatively invariant distributions and a matrix realization of homogeneous cones obtained by the second author [see ibid. 52, No. 1, 161–186 (2000; Zbl 0954.43003)].
MSC:
| 62H05 | Characterization and structure theory for multivariate probability distributions; copulas |
| 15B48 | Positive matrices and their generalizations; cones of matrices |
| 43A99 | Abstract harmonic analysis |
Citations:
Zbl 0954.43003References:
| [1] | S. A. Andersson and G. G. Wojnar, Wishart distributions on homogeneous cones, J. Theoret. Probab., 17 (2004), 781-818. · Zbl 1058.62044 · doi:10.1007/s10959-004-0576-z |
| [2] | M. S. Bartlett, On the theory of statistical regression, Proc. Royal Soc. Edinburgh, 53 (1933), 260-283. · Zbl 0008.02402 |
| [3] | I. Boutouria, Characterization of the Wishart distributions on homogeneous cones, C. R. Math. Acad. Sci. Paris, 341 (2005), 43-48. · Zbl 1065.62095 · doi:10.1016/j.crma.2005.05.022 |
| [4] | P. Diaconis and D. Ylvisaker, Conjugate priors for exponential families, Ann. Statist., 7 (1979), 269-281. · Zbl 0405.62011 · doi:10.1214/aos/1176344611 |
| [5] | J. Faraut and A. Korányi, Analysis on Symmetric Cones, Oxford Math. Monogr., Clarendon Press, 1994. · Zbl 0841.43002 |
| [6] | L. Gå rding, The solution of Cauchy’s problem for two totally hyperbolic linear differential equations by means of Riesz integrals, Ann. of Math. (2), 48 (1947), 785-826. · Zbl 0029.21601 · doi:10.2307/1969381 |
| [7] | S. G. Gindikin, Analysis in homogeneous domains, Russian Math. Surveys, 19 (1964), 1-89. · Zbl 0144.08101 |
| [8] | S. G. Gindikin, Invariant generalized functions in homogeneous domains, Funct. Anal. Appl., 9 (1975), 50-52. · Zbl 0332.32022 · doi:10.1007/BF01078179 |
| [9] | P. Graczyk, G. Letac and H. Massam, The complex Wishart distribution and the symmetric group, Ann. Statist., 31 (2003), 287-309. · Zbl 1019.62047 · doi:10.1214/aos/1046294466 |
| [10] | P. Graczyk, G. Letac and H. Massam, The hyperoctahedral group, symmetric group representations and the moments of the real Wishart distribution, J. Theoret. Probab., 18 (2005), 1-42. · Zbl 1067.60003 · doi:10.1007/s10959-004-0579-9 |
| [11] | A. Hassairi and S. Lajmi, Riesz exponential families on symmetric cones, J. Theoret. Probab., 14 (2001), 927-948. · Zbl 0984.60022 · doi:10.1023/A:1012592618872 |
| [12] | H. Ishi, Positive Riesz distributions on homogeneous cones, J. Math. Soc. Japan, 52 (2000), 161-186. · Zbl 0954.43003 · doi:10.2969/jmsj/05210161 |
| [13] | H. Ishi, Basic relative invariants associated to homogeneous cones and applications, J. Lie Theory, 11 (2001), 155-171. · Zbl 0976.43005 |
| [14] | H. Ishi, The gradient maps associated to certain non-homogeneous cones, Proc. Japan Acad. Ser. A Math. Sci., 81 (2005), 44-46. · Zbl 1086.52501 · doi:10.3792/pjaa.81.44 |
| [15] | H. Ishi, On symplectic representations of normal \(j\)-algebras and their application to Xu’s realizations of Siegel domains, Differential Geom. Appl., 24 (2006), 588-612. · Zbl 1156.17300 · doi:10.1016/j.difgeo.2006.02.001 |
| [16] | H. Ishi, Representation of clans and homogeneous cones, Vestnik Tambov University, 16 (2011), 1669-1675. |
| [17] | M. Katori and H. Tanemura, Symmetry of matrix-valued stochastic processes and noncolliding diffusion particle systems, J. Math. Phys., 45 (2004), 3058-3085. · Zbl 1071.82045 · doi:10.1063/1.1765215 |
| [18] | A. M. Kshirsagar, Bartlett decomposition and Wishart distribution, Ann. Math. Statist., 30 (1959), 239-241. · Zbl 0094.33202 · doi:10.1214/aoms/1177706379 |
| [19] | V. P. Leonov and A. N. Sirjaev, On a method of semi-invariants, Theor. Probability Appl., 4 (1959), 319-329. · Zbl 0087.33701 · doi:10.1137/1104031 |
| [20] | G. Letac and H. Massam, Wishart distributions for decomposable graphs, Ann. Statist., 35 (2007), 1278-1323. · Zbl 1194.62078 · doi:10.1214/009053606000001235 |
| [21] | R. J. Muirhead, Aspects of Multivariate Statistical Theory, John Wiley & Sons, New York, 1982. · Zbl 0556.62028 |
| [22] | M. Riesz, L’intégrale de Riemann-Liouville et le problème de Cauchy, Acta Math., 81 (1949), 1-223. · Zbl 0033.27601 · doi:10.1007/BF02395016 |
| [23] | O. S. Rothaus, The construction of homogeneous convex cones, Ann. of Math. (2), 83 , 358-376, Correction: ibid , 87 (1968), 399. · Zbl 0138.43302 · doi:10.2307/1970436 |
| [24] | È. B. Vinberg, Homogeneous cones, Soviet Math. Dokl., 1 (1960), 787-790. · Zbl 0143.05203 |
| [25] | È. B. Vinberg, The theory of convex homogeneous cones, Trans. Amer. Math. Soc., 12 (1963), 340-403. · Zbl 0138.43301 |
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