Bernoulli

  • Bernoulli
  • Volume 19, Number 2 (2013), 548-598.

Orthogonal polynomial kernels and canonical correlations for Dirichlet measures

Robert C. Griffiths and Dario Spanò

Full-text: Open access

Abstract

We consider a multivariate version of the so-called Lancaster problem of characterizing canonical correlation coefficients of symmetric bivariate distributions with identical marginals and orthogonal polynomial expansions. The marginal distributions examined in this paper are the Dirichlet and the Dirichlet multinomial distribution, respectively, on the continuous and the $N$-discrete $d$-dimensional simplex. Their infinite-dimensional limit distributions, respectively, the Poisson–Dirichlet distribution and Ewens’s sampling formula, are considered as well. We study, in particular, the possibility of mapping canonical correlations on the $d$-dimensional continuous simplex (i) to canonical correlation sequences on the $d+1$-dimensional simplex and/or (ii) to canonical correlations on the discrete simplex, and vice versa. Driven by this motivation, the first half of the paper is devoted to providing a full characterization and probabilistic interpretation of $n$-orthogonal polynomial kernels (i.e., sums of products of orthogonal polynomials of the same degree $n$) with respect to the mentioned marginal distributions. We establish several identities and some integral representations which are multivariate extensions of important results known for the case $d=2$ since the 1970s. These results, along with a common interpretation of the mentioned kernels in terms of dependent Pólya urns, are shown to be key features leading to several non-trivial solutions to Lancaster’s problem, many of which can be extended naturally to the limit as $d\rightarrow\infty$.

Article information

Source
Bernoulli, Volume 19, Number 2 (2013), 548-598.

Dates
First available in Project Euclid: 13 March 2013

Permanent link to this document
https://projecteuclid.org/euclid.bj/1363192038

Digital Object Identifier
doi:10.3150/11-BEJ403

Mathematical Reviews number (MathSciNet)
MR3037164

Zentralblatt MATH identifier
1281.60015

Keywords
canonical correlations Dirichlet distribution Dirichlet-multinomial distribution Ewens’s sampling formula Hahn Jacobi Lancaster’s problem multivariate orthogonal polynomials orthogonal polynomial kernels Poisson–Dirichlet distribution Pólya urns positive-definite sequences

Citation

Griffiths, Robert C.; Spanò, Dario. Orthogonal polynomial kernels and canonical correlations for Dirichlet measures. Bernoulli 19 (2013), no. 2, 548--598. doi:10.3150/11-BEJ403. https://projecteuclid.org/euclid.bj/1363192038


Export citation

References

  • [1] Abramowitz, M. and Stegun, I.A. (1964). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. National Bureau of Standards Applied Mathematics Series 55. New York: Dover.
  • [2] Andrews, G.E., Askey, R. and Roy, R. (1999). Special Functions. Encyclopedia of Mathematics and Its Applications 71. Cambridge: Cambridge Univ. Press.
  • [3] Bochner, S. (1954). Positive zonal functions on spheres. Proc. Nat. Acad. Sci. U.S.A. 40 1141–1147.
  • [4] Diaconis, P., Khare, K. and Saloff-Coste, L. (2008). Gibbs sampling, exponential families and orthogonal polynomials (with comments and a rejoinder by the authors). Statist. Sci. 23 151–178.
  • [5] Dunkl, C.F. and Xu, Y. (2001). Orthogonal Polynomials of Several Variables. Encyclopedia of Mathematics and Its Applications 81. Cambridge: Cambridge Univ. Press.
  • [6] Eagleson, G.K. (1964). Polynomial expansions of bivariate distributions. Ann. Math. Statist. 35 1208–1215.
  • [7] Eagleson, G.K. and Lancaster, H.O. (1967). The regression system of sums with random elements in common. Austral. J. Statist. 9 119–125.
  • [8] Gasper, G. (1972). Banach algebras for Jacobi series and positivity of a kernel. Ann. of Math. (2) 95 261–280.
  • [9] Gasper, G. (1973). Nonnegativity of a discrete Poisson kernel for the Hahn polynomials. J. Math. Anal. Appl. 42 438–451. Collection of articles dedicated to Salomon Bochner.
  • [10] Griffiths, R.C. (1979). On the distribution of allele frequencies in a diffusion model. Theoret. Population Biol. 15 140–158.
  • [11] Griffiths, R.C. (1979). A transition density expansion for a multi-allele diffusion model. Adv. in Appl. Probab. 11 310–325.
  • [12] Griffiths, R.C. (1980). Lines of descent in the diffusion approximation of neutral Wright–Fisher models. Theoret. Population Biol. 17 37–50.
  • [13] Griffiths, R.C. (2006). Coalescent lineage distributions. Adv. in Appl. Probab. 38 405–429.
  • [14] Griffiths, R.C. and Spanó, D. (2010). Diffusion processes and coalescent trees. In Probability and Mathematical Genetics. London Mathematical Society Lecture Note Series 378 358–379. Cambridge: Cambridge Univ. Press.
  • [15] Griffiths, R.C. and Spanò, D. (2011). Multivariate Jacobi and Laguerre polynomials, infinite-dimensional extensions, and their probabilistic connections with multivariate Hahn and Meixner polynomials. Bernoulli 17 1095–1125.
  • [16] Ismail, M.E.H. (2005). Classical and Quantum Orthogonal Polynomials in One Variable. Encyclopedia of Mathematics and Its Applications 98. Cambridge: Cambridge Univ. Press. With two chapters by Walter Van Assche, with a foreword by Richard A. Askey.
  • [17] Karlin, S. and McGregor, J. (1975). Linear growth models with many types and multidimensional Hahn polynomials. In Theory and Application of Special Functions (Proc. Advanced Sem., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1975) 261–288. New York: Academic Press.
  • [18] Khare, K. and Zhou, H. (2009). Rates of convergence of some multivariate Markov chains with polynomial eigenfunctions. Ann. Appl. Probab. 19 737–777.
  • [19] Kingman, J.F.C., Taylor, S.J., Hawkes, A.G., Walker, A.M., Cox, D.R., Smith, A.F.M., Hill, B.M., Burville, P.J. and Leonard, T. (1975). Random discrete distribution. J. Roy. Statist. Soc. Ser. B 37 1–22. With a discussion by S.J. Taylor, A.G. Hawkes, A.M. Walker, D.R. Cox, A.F.M. Smith, B.M. Hill, P.J. Burville, T. Leonard and a reply by the author.
  • [20] Koornwinder, T. (1974). Jacobi polynomials. II. An analytic proof of the product formula. SIAM J. Math. Anal. 5 125–137.
  • [21] Koornwinder, T.H. and Schwartz, A.L. (1997). Product formulas and associated hypergroups for orthogonal polynomials on the simplex and on a parabolic biangle. Constr. Approx. 13 537–567.
  • [22] Lancaster, H.O. (1958). The structure of bivariate distributions. Ann. Math. Statist. 29 719–736.
  • [23] Muliere, P., Secchi, P. and Walker, S. (2005). Partially exchangeable processes indexed by the vertices of a $k$-tree constructed via reinforcement. Stochastic Process. Appl. 115 661–677.
  • [24] Peccati, G. (2008). Multiple integral representation for functionals of Dirichlet processes. Bernoulli 14 91–124.
  • [25] Rosengren, H. (1999). Multivariable orthogonal polynomials and coupling coefficients for discrete series representations. SIAM J. Math. Anal. 30 232–272 (electronic).
  • [26] Waldron, S. (2006). On the Bernstein–Bézier form of Jacobi polynomials on a simplex. J. Approx. Theory 140 86–99.
  • [27] Watterson, G.A. (1984). Lines of descent and the coalescent. Theoret. Population Biol. 26 77–92.