Quadratic harnesses, 𝑞-commutations, and orthogonal martingale polynomials

Bryc, Włodzimierz; Matysiak, Wojciech; Wesołowski, Jacek

doi:10.1090/S0002-9947-07-04194-3

Quadratic harnesses, $q$-commutations, and orthogonal martingale polynomials
HTML articles powered by AMS MathViewer

by Włodzimierz Bryc, Wojciech Matysiak and Jacek Wesołowski PDF

Trans. Amer. Math. Soc. 359 (2007), 5449-5483 Request permission

Abstract:

We introduce the quadratic harness condition and show that integrable quadratic harnesses have orthogonal martingale polynomials with a three step recurrence that satisfies a $q$-commutation relation. This implies that quadratic harnesses are essentially determined uniquely by five numerical constants. Explicit recurrences for the orthogonal martingale polynomials are derived in several cases of interest.

References

W. A. Al-Salam and T. S. Chihara, Convolutions of orthonormal polynomials, SIAM J. Math. Anal. 7 (1976), no. 1, 16–28. MR 399537, DOI 10.1137/0507003
Michael Anshelevich, Free martingale polynomials, J. Funct. Anal. 201 (2003), no. 1, 228–261. MR 1986160, DOI 10.1016/S0022-1236(03)00061-2
Nobuhiro Asai, Izumi Kubo, and Hui-Hsiung Kuo, Segal-Bargmann transforms of one-mode interacting Fock spaces associated with Gaussian and Poisson measures, Proc. Amer. Math. Soc. 131 (2003), no. 3, 815–823. MR 1937419, DOI 10.1090/S0002-9939-02-06564-4
Richard Askey and Mourad Ismail, Recurrence relations, continued fractions, and orthogonal polynomials, Mem. Amer. Math. Soc. 49 (1984), no. 300, iv+108. MR 743545, DOI 10.1090/memo/0300
Richard Askey and James Wilson, Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials, Mem. Amer. Math. Soc. 54 (1985), no. 319, iv+55. MR 783216, DOI 10.1090/memo/0319
Marek Bożejko and Włodzimierz Bryc, On a class of free Lévy laws related to a regression problem, J. Funct. Anal. 236 (2006), no. 1, 59–77. MR 2227129, DOI 10.1016/j.jfa.2005.09.010
Marek Bożejko, Burkhard Kümmerer, and Roland Speicher, $q$-Gaussian processes: non-commutative and classical aspects, Comm. Math. Phys. 185 (1997), no. 1, 129–154. MR 1463036, DOI 10.1007/s002200050084
Marek Bożejko and Janusz Wysoczański, Remarks on $t$-transformations of measures and convolutions, Ann. Inst. H. Poincaré Probab. Statist. 37 (2001), no. 6, 737–761 (English, with English and French summaries). MR 1863276, DOI 10.1016/S0246-0203(01)01084-6
Włodzimierz Bryc, Some remarks on random vectors with nice enough behaviour of conditional moments, Bull. Polish Acad. Sci. Math. 33 (1985), no. 11-12, 677–684 (1986) (English, with Russian summary). MR 849420
Włodzimierz Bryc, Wojciech Matysiak, and Jacek Wesołowski. The bi-Poisson process: A quadratic harness. Ann. Probab., to appear.
Włodzimierz Bryc and Agnieszka Plucińska, A characterization of infinite Gaussian sequences by conditional moments, Sankhyā Ser. A 47 (1985), no. 2, 166–173. MR 844017
Włodzimierz Bryc and Jacek Wesołowski. Bi-Poisson process. Infin. Dimens. Anal. Quantum Probab. Related. Top., to appear.
Włodzimierz Bryc and Jacek Wesołowski, The classical bi-Poisson process: an invertible quadratic harness, Statist. Probab. Lett. 76 (2006), no. 15, 1664–1674. MR 2248855, DOI 10.1016/j.spl.2006.04.050
Włodzimierz Bryc and Jacek Wesołowski, Conditional moments of $q$-Meixner processes, Probab. Theory Related Fields 131 (2005), no. 3, 415–441. MR 2123251, DOI 10.1007/s00440-004-0379-2
T. S. Chihara, An introduction to orthogonal polynomials, Mathematics and its Applications, Vol. 13, Gordon and Breach Science Publishers, New York-London-Paris, 1978. MR 0481884
B. Derrida, M. R. Evans, V. Hakim, and V. Pasquier, Exact solution of a $1$D asymmetric exclusion model using a matrix formulation, J. Phys. A 26 (1993), no. 7, 1493–1517. MR 1219679
M. Dozzi, Two-parameter harnesses and the Wiener process, Z. Wahrsch. Verw. Gebiete 56 (1981), no. 4, 507–514. MR 621661, DOI 10.1007/BF00531429
Charles F. Dunkl and Yuan Xu, Orthogonal polynomials of several variables, Encyclopedia of Mathematics and its Applications, vol. 81, Cambridge University Press, Cambridge, 2001. MR 1827871, DOI 10.1017/CBO9780511565717
Fabian H. L. Essler and Vladimir Rittenberg, Representations of the quadratic algebra and partially asymmetric diffusion with open boundaries, J. Phys. A 29 (1996), no. 13, 3375–3407. MR 1400161, DOI 10.1088/0305-4470/29/13/013
Philip Feinsilver, Lie algebras and recurrence relations. III. $q$-analogs and quantized algebras, Acta Appl. Math. 19 (1990), no. 3, 207–251. MR 1077860, DOI 10.1007/BF01321858
U. Frisch and R. Bourret, Parastochastics, J. Mathematical Phys. 11 (1970), 364–390. MR 260352, DOI 10.1063/1.1665149
Léonard Gallardo and Marc Yor, Some new examples of Markov processes which enjoy the time-inversion property, Probab. Theory Related Fields 132 (2005), no. 1, 150–162. MR 2136870, DOI 10.1007/s00440-004-0399-y
J. M. Hammersley, Harnesses, Proc. Fifth Berkeley Sympos. Mathematical Statistics and Probability (Berkeley, Calif., 1965/66) Univ. California Press, Berkeley, Calif., 1967, pp. 89–117. MR 0224144
Mourad E. H. Ismail and Dennis Stanton, $q$-integral and moment representations for $q$-orthogonal polynomials, Canad. J. Math. 54 (2002), no. 4, 709–735. MR 1913916, DOI 10.4153/CJM-2002-027-2
Christian Kassel, Quantum groups, Graduate Texts in Mathematics, vol. 155, Springer-Verlag, New York, 1995. MR 1321145, DOI 10.1007/978-1-4612-0783-2
Anna Dorota Krystek and Łukasz Jan Wojakowski, Associative convolutions arising from conditionally free convolution, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 8 (2005), no. 3, 515–545. MR 2172313, DOI 10.1142/S0219025705002104
I. G. Macdonald, Affine Hecke algebras and orthogonal polynomials, Cambridge Tracts in Mathematics, vol. 157, Cambridge University Press, Cambridge, 2003. MR 1976581, DOI 10.1017/CBO9780511542824
Roger Mansuy and Marc Yor, Harnesses, Lévy bridges and Monsieur Jourdain, Stochastic Process. Appl. 115 (2005), no. 2, 329–338. MR 2111197, DOI 10.1016/j.spa.2004.09.001
Masatoshi Noumi and Jasper V. Stokman, Askey-Wilson polynomials: an affine Hecke algebra approach, Laredo Lectures on Orthogonal Polynomials and Special Functions, Adv. Theory Spec. Funct. Orthogonal Polynomials, Nova Sci. Publ., Hauppauge, NY, 2004, pp. 111–144. MR 2085854
Eugene A. Pechersky Pablo A. Ferrari, and Beat M. Niederhauser. Harness processes and non-homogeneous crystals. arXiv:math.PR/0409301, 2004.
A. Perelomov, Generalized coherent states and their applications, Texts and Monographs in Physics, Springer-Verlag, Berlin, 1986. MR 858831, DOI 10.1007/978-3-642-61629-7
André Ronveaux and Walter Van Assche, Upward extension of the Jacobi matrix for orthogonal polynomials, J. Approx. Theory 86 (1996), no. 3, 335–357. MR 1405986, DOI 10.1006/jath.1996.0074
Gian-Carlo Rota (ed.), Finite operator calculus, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1975. With the collaboration of P. Doubilet, C. Greene, D. Kahaner, A. Odlyzko and R. Stanley. MR 0379213
Gabriela Sansigre and Galliano Valent, A large family of semi-classical polynomials: the perturbed Chebyshev, Proceedings of the Fourth International Symposium on Orthogonal Polynomials and their Applications (Evian-Les-Bains, 1992), 1995, pp. 271–281. MR 1340942, DOI 10.1016/0377-0427(93)E0251-G
Wim Schoutens, Stochastic processes and orthogonal polynomials, Lecture Notes in Statistics, vol. 146, Springer-Verlag, New York, 2000. MR 1761401, DOI 10.1007/978-1-4612-1170-9
Masaru Uchiyama, Tomohiro Sasamoto, and Miki Wadati, Asymmetric simple exclusion process with open boundaries and Askey-Wilson polynomials, J. Phys. A 37 (2004), no. 18, 4985–5002. MR 2065218, DOI 10.1088/0305-4470/37/18/006
Hans van Leeuwen and Hans Maassen, A $q$ deformation of the Gauss distribution, J. Math. Phys. 36 (1995), no. 9, 4743–4756. MR 1347109, DOI 10.1063/1.530917
A. M. Vershik, Algebras with quadratic relations [translation of Spectral theory of operators and infinite-dimensional analysis (Russian), 32–57, ii, Akad. Nauk. Ukrain. SSR, Inst. Mat., Kiev, 1984; MR0817217 (87m:16044)], Selecta Math. Soviet. 11 (1992), no. 4, 293–315. Selected translations. MR 1206295
Jacek Wesołowski, Stochastic processes with linear conditional expectation and quadratic conditional variance, Probab. Math. Statist. 14 (1993), no. 1, 33–44. MR 1267516
David Williams, Some basic theorems on harnesses, Stochastic analysis (a tribute to the memory of Rollo Davidson), Wiley, London, 1973, pp. 349–363. MR 0362565
Zhan Gong Zhou, Two-parameter harnesses and the generalized Brownian sheet, Natur. Sci. J. Xiangtan Univ. 14 (1992), no. 2, 111–115 (Chinese, with English and Chinese summaries). MR 1184355
Xing Wu Zhuang, The generalized Brownian sheet and two-parameter harnesses, Fujian Shifan Daxue Xuebao Ziran Kexue Ban 4 (1988), no. 4, 1–9 (Chinese, with English summary). MR 1041927