×

zbMATH — the first resource for mathematics

Quadratic harnesses, \(q\)-commutations, and orthogonal martingale polynomials. (English) Zbl 1129.60068
40 years ago Hammersley introduced the concept of harnesses on \(\mathbb{R}^n\) as probabilistic models of longrange misorientation in the crystalline structure of metals. An important subclass are the so-called quadratic harnesses. The main focus of the paper is on uniqueness and properties. More precisely, the authors show that quadratic harnesses are described via five numerical constants, which, under certain integrability conditions, in fact, determine the process. Another topic concerns properties of martingale polynomials associated with quadratic harnesses with finite moments of all orders processes. Some glimps into ongoing further work and future possible work are also given.

MSC:
60J25 Continuous-time Markov processes on general state spaces
46L53 Noncommutative probability and statistics
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] W. A. Al-Salam and T. S. Chihara, Convolutions of orthonormal polynomials, SIAM J. Math. Anal. 7 (1976), no. 1, 16 – 28. · Zbl 0323.33007
[2] Michael Anshelevich, Free martingale polynomials, J. Funct. Anal. 201 (2003), no. 1, 228 – 261. · Zbl 1033.46050
[3] 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. · Zbl 1028.46038
[4] Richard Askey and Mourad Ismail, Recurrence relations, continued fractions, and orthogonal polynomials, Mem. Amer. Math. Soc. 49 (1984), no. 300, iv+108. · Zbl 0548.33001
[5] Richard Askey and James Wilson, Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials, Mem. Amer. Math. Soc. 54 (1985), no. 319, iv+55. · Zbl 0572.33012
[6] 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. · Zbl 1110.46040
[7] Marek Bożejko, Burkhard Kümmerer, and Roland Speicher, \?-Gaussian processes: non-commutative and classical aspects, Comm. Math. Phys. 185 (1997), no. 1, 129 – 154. · Zbl 0873.60087
[8] Marek Bożejko and Janusz Wysoczański, Remarks on \?-transformations of measures and convolutions, Ann. Inst. H. Poincaré Probab. Statist. 37 (2001), no. 6, 737 – 761 (English, with English and French summaries). · Zbl 0995.60004
[9] 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). · Zbl 0612.60020
[10] W\lodzimierz Bryc, Wojciech Matysiak, and Jacek Weso\lowski. The bi-Poisson process: A quadratic harness. Ann. Probab., to appear. · Zbl 1137.60036
[11] Włodzimierz Bryc and Agnieszka Plucińska, A characterization of infinite Gaussian sequences by conditional moments, Sankhyā Ser. A 47 (1985), no. 2, 166 – 173. · Zbl 0609.62021
[12] W\lodzimierz Bryc and Jacek Weso\lowski. Bi-Poisson process. Infin. Dimens. Anal. Quantum Probab. Related. Top., to appear.
[13] Włodzimierz Bryc and Jacek Wesołowski, The classical bi-Poisson process: an invertible quadratic harness, Statist. Probab. Lett. 76 (2006), no. 15, 1664 – 1674. · Zbl 1105.60053
[14] Włodzimierz Bryc and Jacek Wesołowski, Conditional moments of \?-Meixner processes, Probab. Theory Related Fields 131 (2005), no. 3, 415 – 441. · Zbl 1118.60065
[15] T. S. Chihara, An introduction to orthogonal polynomials, Gordon and Breach Science Publishers, New York-London-Paris, 1978. Mathematics and its Applications, Vol. 13. · Zbl 0389.33008
[16] B. Derrida, M. R. Evans, V. Hakim, and V. Pasquier, Exact solution of a 1D asymmetric exclusion model using a matrix formulation, J. Phys. A 26 (1993), no. 7, 1493 – 1517. · Zbl 0772.60096
[17] M. Dozzi, Two-parameter harnesses and the Wiener process, Z. Wahrsch. Verw. Gebiete 56 (1981), no. 4, 507 – 514. · Zbl 0456.60047
[18] Charles F. Dunkl and Yuan Xu, Orthogonal polynomials of several variables, Encyclopedia of Mathematics and its Applications, vol. 81, Cambridge University Press, Cambridge, 2001. · Zbl 0964.33001
[19] 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. · Zbl 0902.60088
[20] Philip Feinsilver, Lie algebras and recurrence relations. III. \?-analogs and quantized algebras, Acta Appl. Math. 19 (1990), no. 3, 207 – 251. · Zbl 0741.33015
[21] U. Frisch and R. Bourret, Parastochastics, J. Mathematical Phys. 11 (1970), 364 – 390. · Zbl 0187.25902
[22] 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. · Zbl 1087.60058
[23] 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.
[24] Mourad E. H. Ismail and Dennis Stanton, \?-integral and moment representations for \?-orthogonal polynomials, Canad. J. Math. 54 (2002), no. 4, 709 – 735. · Zbl 1009.33015
[25] Christian Kassel, Quantum groups, Graduate Texts in Mathematics, vol. 155, Springer-Verlag, New York, 1995. · Zbl 0808.17003
[26] 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. · Zbl 1087.46042
[27] I. G. Macdonald, Affine Hecke algebras and orthogonal polynomials, Cambridge Tracts in Mathematics, vol. 157, Cambridge University Press, Cambridge, 2003. · Zbl 1024.33001
[28] Roger Mansuy and Marc Yor, Harnesses, Lévy bridges and Monsieur Jourdain, Stochastic Process. Appl. 115 (2005), no. 2, 329 – 338. · Zbl 1070.60041
[29] 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. · Zbl 1096.33008
[30] Eugene A. Pechersky Pablo A. Ferrari, and Beat M. Niederhauser. Harness processes and non-homogeneous crystals. arXiv:math.PR/0409301, 2004. · Zbl 1206.82118
[31] A. Perelomov, Generalized coherent states and their applications, Texts and Monographs in Physics, Springer-Verlag, Berlin, 1986. · Zbl 0605.22013
[32] André Ronveaux and Walter Van Assche, Upward extension of the Jacobi matrix for orthogonal polynomials, J. Approx. Theory 86 (1996), no. 3, 335 – 357. · Zbl 0858.42015
[33] Gian-Carlo Rota , 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. · Zbl 0328.05007
[34] 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. · Zbl 0831.42016
[35] Wim Schoutens, Stochastic processes and orthogonal polynomials, Lecture Notes in Statistics, vol. 146, Springer-Verlag, New York, 2000. · Zbl 0960.60076
[36] 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. · Zbl 1047.82019
[37] Hans van Leeuwen and Hans Maassen, A \? deformation of the Gauss distribution, J. Math. Phys. 36 (1995), no. 9, 4743 – 4756. · Zbl 0841.60089
[38] 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.
[39] Jacek Wesołowski, Stochastic processes with linear conditional expectation and quadratic conditional variance, Probab. Math. Statist. 14 (1993), no. 1, 33 – 44. · Zbl 0803.60073
[40] David Williams, Some basic theorems on harnesses, Stochastic analysis (a tribute to the memory of Rollo Davidson), Wiley, London, 1973, pp. 349 – 363.
[41] 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). · Zbl 0760.60052
[42] 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). · Zbl 1382.60108
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.