The Annals of Probability

Askey–Wilson polynomials, quadratic harnesses and martingales

Włodek Bryc and Jacek Wesołowski

Full-text: Open access

Abstract

We use orthogonality measures of Askey–Wilson polynomials to construct Markov processes with linear regressions and quadratic conditional variances. Askey–Wilson polynomials are orthogonal martingale polynomials for these processes.

Article information

Source
Ann. Probab., Volume 38, Number 3 (2010), 1221-1262.

Dates
First available in Project Euclid: 2 June 2010

Permanent link to this document
https://projecteuclid.org/euclid.aop/1275486192

Digital Object Identifier
doi:10.1214/09-AOP503

Mathematical Reviews number (MathSciNet)
MR2674998

Zentralblatt MATH identifier
1201.60077

Subjects
Primary: 60J25: Continuous-time Markov processes on general state spaces
Secondary: 46L53: Noncommutative probability and statistics

Keywords
Quadratic conditional variances harnesses orthogonal martingale polynomials hypergeometric orthogonal polynomials

Citation

Bryc, Włodek; Wesołowski, Jacek. Askey–Wilson polynomials, quadratic harnesses and martingales. Ann. Probab. 38 (2010), no. 3, 1221--1262. doi:10.1214/09-AOP503. https://projecteuclid.org/euclid.aop/1275486192


Export citation

References

  • [1] Anshelevich, M. (2003). Free martingale polynomials. J. Funct. Anal. 201 228–261.
  • [2] Anshelevich, M. (2004). Appell polynomials and their relatives. Int. Math. Res. Not. IMRN 65 3469–3531.
  • [3] Askey, R. and Wilson, J. (1979). A set of orthogonal polynomials that generalize the Racah coefficients or 6−j symbols. SIAM J. Math. Anal. 10 1008–1016.
  • [4] Askey, R. and Wilson, J. (1985). Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials. Mem. Amer. Math. Soc. 54 iv+55.
  • [5] Bakry, D. and Mazet, O. (2003). Characterization of Markov semigroups on ℝ associated to some families of orthogonal polynomials. In Séminaire de Probabilités XXXVII. Lecture Notes in Math. 1832 60–80. Springer, Berlin.
  • [6] Biane, P. (1998). Processes with free increments. Math. Z. 227 143–174.
  • [7] Bożejko, M. and Bryc, W. (2006). On a class of free Lévy laws related to a regression problem. J. Funct. Anal. 236 59–77.
  • [8] Bryc, W., Matysiak, W. and Wesołowski, J. (2007). Quadratic harnesses, q-commutations, and orthogonal martingale polynomials. Trans. Amer. Math. Soc. 359 5449–5483.
  • [9] Bryc, W., Matysiak, W. and Wesołowski, J. (2008). The bi-Poisson process: A quadratic harness. Ann. Probab. 36 623–646.
  • [10] Bryc, W. and Wesołowski, J. (2005). Conditional moments of q-Meixner processes. Probab. Theory Related Fields 131 415–441.
  • [11] Bryc, W. and Wesołowski, J. (2007). Bi-Poisson process. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 10 277–291.
  • [12] Diaconis, P., Khare, K. and Saloff-Coste, L. (2008). Rejoinder: Gibbs sampling, exponential families and orthogonal polynomials. Statist. Sci. 23 196–200.
  • [13] Dunkl, C. F. and Xu, Y. (2001). Orthogonal Polynomials of Several Variables. Encyclopedia of Mathematics and Its Applications 81. Cambridge Univ. Press, Cambridge.
  • [14] Feinsilver, P. (1986). Some classes of orthogonal polynomials associated with martingales. Proc. Amer. Math. Soc. 98 298–302.
  • [15] Hammersley, J. M. (1967). Harnesses. In Proc. Fifth Berkeley Symp. Math. Statist. Probab. III: Physical Sciences 89–117. Univ. California Press, Berkeley, CA.
  • [16] Hiai, F. and Petz, D. (2000). The Semicircle Law, Free Random Variables and Entropy. Mathematical Surveys and Monographs 77. Amer. Math. Soc., Providence, RI.
  • [17] Ismail, M. E. H. (2005). Classical and Quantum Orthogonal Polynomials in One Variable. Encyclopedia of Mathematics and Its Applications 98. Cambridge Univ. Press, Cambridge.
  • [18] Koekoek, R. and Swarttouw, R. F. (1998). The Askey scheme of hypergeometric orthogonal polynomials and its q-analogue. Report 98-17, Delft Univ. Technology. Available at http://fa.its.tudelft.nl/~koekoek/askey.html.
  • [19] Lytvynov, E. (2003). Polynomials of Meixner’s type in infinite dimensions—Jacobi fields and orthogonality measures. J. Funct. Anal. 200 118–149.
  • [20] Mansuy, R. and Yor, M. (2005). Harnesses, Lévy bridges and Monsieur Jourdain. Stochastic Process. Appl. 115 329–338.
  • [21] Nassrallah, B. and Rahman, M. (1985). Projection formulas, a reproducing kernel and a generating function for q-Wilson polynomials. SIAM J. Math. Anal. 16 186–197.
  • [22] Noumi, M. and Stokman, J. V. (2004). Askey–Wilson polynomials: An affine Hecke algebra approach. In Laredo Lectures on Orthogonal Polynomials and Special Functions. Adv. Theory Spec. Funct. Orthogonal Polynomials 111–144. Nova Sci. Publ., Hauppauge, NY.
  • [23] Nualart, D. and Schoutens, W. (2000). Chaotic and predictable representations for Lévy processes. Stochastic Process. Appl. 90 109–122.
  • [24] Saitoh, N. and Yoshida, H. (2000). A q-deformed Poisson distribution based on orthogonal polynomials. J. Phys. A 33 1435–1444.
  • [25] Schoutens, W. (2000). Stochastic Processes and Orthogonal Polynomials. Lecture Notes in Statistics 146. Springer, New York.
  • [26] Schoutens, W. and Teugels, J. L. (1998). Lévy processes, polynomials and martingales. Comm. Statist. Stochastic Models 14 335–349.
  • [27] Solé, J. L. and Utzet, F. (2008). On the orthogonal polynomials associated with a Lévy process. Ann. Probab. 36 765–795.
  • [28] Solé, J. L. and Utzet, F. (2008). Time–space harmonic polynomials relative to a Lévy process. Bernoulli 14 1–13.
  • [29] Stokman, J. V. (1997). Multivariable BC type Askey-Wilson polynomials with partly discrete orthogonality measure. Ramanujan J. 1 275–297.
  • [30] Stokman, J. V. and Koornwinder, T. H. (1998). On some limit cases of Askey–Wilson polynomials. J. Approx. Theory 95 310–330.
  • [31] Szegö, G. (1939). Orthogonal Polynomials. American Mathematical Society Colloquium Publications 23. Amer. Math. Soc., New York.
  • [32] Uchiyama, M., Sasamoto, T. and Wadati, M. (2004). Asymmetric simple exclusion process with open boundaries and Askey–Wilson polynomials. J. Phys. A 37 4985–5002.
  • [33] Wesołowski, J. (1993). Stochastic processes with linear conditional expectation and quadratic conditional variance. Probab. Math. Statist. 14 33–44.
  • [34] Williams, D. (1973). Some basic theorems on harnesses. In Stochastic Analysis (a Tribute to the Memory of Rollo Davidson) 349–363. Wiley, London.