Transactions of the American Mathematical Society

Author(s): Wlodzimierz Bryc; Wojciech Matysiak; Jacek Wesolowski
Journal: Trans. Amer. Math. Soc. 359 (2007), 5449-5483.
MSC (2000): Primary 60J25; Secondary 46L53
Posted: June 13, 2007
Retrieve article in: PDF

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

-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.

1.: W. A. Al-Salam and T. S. Chihara.
Convolutions of orthonormal polynomials.
SIAM J. Math. Anal., 7(1):16-28, 1976. MR 0399537 (53:3381)
2.: Michael Anshelevich.
Free martingale polynomials.
Journal of Functional Analysis, (201):228-261, 2003.
arXiv:math.CO/0112194. MR 1986160 (2004f:46079)
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(3):815-823, 2003. MR 1937419 (2003j:46099)
4.: Richard Askey and Mourad Ismail.
Recurrence relations, continued fractions, and orthogonal polynomials.
Mem. Amer. Math. Soc., 49(300):iv+108, 1984. MR 743545 (85g:33008)
5.: Richard Askey and James Wilson.
Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials.
Mem. Amer. Math. Soc., 54(319):iv+55, 1985. MR 783216 (87a:05023)
6.: Marek Bozejko and W $\l$ odzimierz Bryc.
On a class of free Lévy laws related to a regression problem. J. Funct. Anal. 236:59-77, 2006. MR 2227129 (2007a:46071)
7.: Marek Bozejko, Burkhard Kümmerer, and Roland Speicher.
-Gaussian processes: non-commutative and classical aspects.
Comm. Math. Phys., 185(1):129-154, 1997. MR 1463036 (98h:81053)
8.: Marek Bozejko and Janusz Wysoczanski.
Remarks on -transformations of measures and convolutions.
Ann. Inst. H. Poincaré Probab. Statist., 37(6):737-761, 2001. MR 1863276 (2002i:60005)
9.: W $\l$ odzimierz Bryc.
Some remarks on random vectors with nice enough behaviour of conditional moments.
Bull. Polish Acad. Sci., 33:677-683, 1985. MR 849420 (88d:60021)
10.: W $\l$ odzimierz Bryc, Wojciech Matysiak, and Jacek Weso $\l$ owski.
The bi-Poisson process: A quadratic harness.
Ann. Probab., to appear.
11.: W $\l$ odzimierz Bryc and Agnieszka Plucinska.
A characterization of infinite Gaussian sequences by conditional moments.
Sankhya A, 47:166-173, 1985. MR 844017 (87k:62023)
12.: W $\l$ odzimierz Bryc and Jacek Weso $\l$ owski.
Bi-Poisson process.
Infin. Dimens. Anal. Quantum Probab. Related. Top., to appear.
13.: W $\l$ odzimierz Bryc and Jacek Weso $\l$ owski.
The classical bi-Poisson process: An invertible quadratic harness. Statist. Probab. Lett. 76:1664-1674, 2006. MR 2248855
14.: W $\l$ odzimierz Bryc and Jacek Weso $\l$ owski.
Conditional moments of -Meixner processes.
Probability Theory Related Fields, 131:415-441, 2005.
arXiv:math.PR/0403016. MR 2123251 (2005k:60233)
15.: T. S. Chihara.
An introduction to orthogonal polynomials.
Gordon and Breach, New York, 1978. MR 0481884 (58:1979)
16.: B. Derrida, M. R. Evans, V. Hakim, and V. Pasquier.
Exact solution of a D asymmetric exclusion model using a matrix formulation.
J. Phys. A, 26(7):1493-1517, 1993. MR 1219679 (94g:60179)
17.: M. Dozzi.
Two-parameter harnesses and the Wiener process.
Z. Wahrsch. Verw. Gebiete, 56(4):507-514, 1981. MR 621661 (82h:60152)
18.: Charles F. Dunkl and Yuan Xu.
Orthogonal polynomials of several variables, volume 81 of Encyclopedia of Mathematics and its Applications.
Cambridge University Press, Cambridge, 2001. MR 1827871 (2002m: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(13):3375-3407, 1996. MR 1400161 (97b:82017)
20.: Philip Feinsilver.
Lie algebras and recurrence relations. III. -analogs and quantized algebras.
Acta Appl. Math., 19(3):207-251, 1990. MR 1077860 (93k:33012)
21.: U. Frisch and R. Bourret.
Parastochastics.
J. Math. Phys., 11(2):364-390, 1970. MR 0260352 (41:4979)
22.: Léonard Gallardo and Marc Yor.
Some new examples of Markov processes which enjoy the time-inversion property.
Probab. Theory Related Fields, 132(1):150-162, 2005. MR 2136870 (2006e:60106)
23.: J. M. Hammersley.
Harnesses.
In Proc. Fifth Berkeley Sympos. Mathematical Statistics and Probability (Berkeley, California, 1965/66), Vol. III: Physical Sciences, pages 89-117. Univ. California Press, Berkeley, California, 1967. MR 0224144 (36:7190)
24.: Mourad E. H. Ismail and Dennis Stanton.
-integral and moment representations for -orthogonal polynomials.
Canad. J. Math., 54(4):709-735, 2002. MR 1913916 (2003k:33024)
25.: Christian Kassel.
Quantum groups, volume 155 of Graduate Texts in Mathematics.
Springer-Verlag, New York, 1995. MR 1321145 (96e:17041)
26.: Anna Krystek and $\L$ ukasz Wojakowski.
Associative convolutions arising from conditionally free convolution.
Infinite Dimensional Analysis, Quantum Probability and Related Topics, 8:515-545, 2005. MR 2172313 (2006g:46103)
27.: I. G. Macdonald.
Affine Hecke algebras and orthogonal polynomials, volume 157 of Cambridge Tracts in Mathematics.
Cambridge University Press, Cambridge, 2003. MR 1976581 (2005b:33021)
28.: R. Mansuy and M. Yor.
Harnesses, Lévy bridges and Monsieur Jourdain.
Stochastic Processes and Their Applications, 115:329-338, February 2005. MR 2111197 (2005m:60104)
29.: Masatoshi Noumi and Jasper V. Stokman.
Askey-Wilson polynomials: An affine Hecke algebra approach.
In Laredo Lectures on Orthogonal Polynomials and Special Functions, Adv. Theory Spec. Funct. Orthogonal Polynomials, pages 111-144. Nova Sci. Publ., Hauppauge, New York, 2004. MR 2085854 (2005h:42057)
30.: Eugene A. Pechersky Pablo A. Ferrari, and Beat M. Niederhauser.
Harness processes and non-homogeneous crystals.
arXiv:math.PR/0409301, 2004.
31.: A. Perelomov.
Generalized coherent states and their applications.
Texts and Monographs in Physics. Springer-Verlag, Berlin, 1986. MR 858831 (87m:22035)
32.: André Ronveaux and Walter Van Assche.
Upward extension of the Jacobi matrix for orthogonal polynomials.
J. Approx. Theory, 86(3):335-357, 1996. MR 1405986 (97k:42054)
33.: Gian-Carlo Rota.
Finite Operator Calculus.
Academic Press, 1975. MR 0379213 (52:119)
34.: Gabriela Sansigre and Galliano Valent.
A large family of semi-classical polynomials: The perturbed Chebyshev.
J. Comput. Appl. Math., 57(1-2):271-281, 1995. MR 1340942 (96f:33021)
35.: Wim Schoutens.
Stochastic processes and orthogonal polynomials, volume 146 of Lecture Notes in Statistics.
Springer-Verlag, New York, 2000. MR 1761401 (2001f:60095)
36.: Masaru Uchiyama, Tomohiro Sasamoto, and Miki Wadati.
Asymmetric simple exclusion process with open boundaries and Askey-Wilson polynomials.
J. Phys. A, 37(18):4985-5002, 2004. MR 2065218 (2006d:82047)
37.: Hans van Leeuwen and Hans Maassen.
A -deformation of the Gauss distribution.
J. Math. Phys., 36(9):4743-4756, 1995. MR 1347109 (97a:81104)
38.: A. M. Vershik.
Algebras with quadratic relations.
Selecta Math. Soviet., 11(4):293-315, 1992.
Selected translations. MR 1206295
39.: Jacek Weso $\l$ owski.
Stochastic processes with linear conditional expectation and quadratic conditional variance.
Probab. Math. Statist., 14:33-44, 1993. MR 1267516 (95b:60061)
40.: David Williams.
Some basic theorems on harnesses.
In Stochastic analysis (a tribute to the memory of Rollo Davidson), pages 349-363. Wiley, London, 1973. MR 0362565 (50:15005)
41.: Zhan Gong Zhou.
Two-parameter harnesses and the generalized Brownian sheet.
Natur. Sci. J. Xiangtan Univ., 14(2):111-115, 1992. MR 1184355 (93h:60129)
42.: Xing Wu Zhuang.
The generalized Brownian sheet and two-parameter harnesses.
Fujian Shifan Daxue Xuebao Ziran Kexue Ban, 4(4):1-9, 1988. MR 1041927 (91e:60158)

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 60J25, 46L53

Wlodzimierz Bryc
Affiliation: Department of Mathematics, University of Cincinnati, P.O. Box 210025, Cincinnati, Ohio 45221--0025
Email: Wlodzimierz.Bryc@UC.edu

Wojciech Matysiak
Affiliation: Faculty of Mathematics and Information Science, Warsaw University of Technology, pl. Politechniki 1, 00-661 Warszawa, Poland
Email: matysiak@mini.pw.edu.pl

Jacek Wesolowski
Affiliation: Faculty of Mathematics and Information Science, Warsaw University of Technology, pl. Politechniki 1, 00-661 Warszawa, Poland
Email: wesolo@alpha.mini.pw.edu.pl

DOI: 10.1090/S0002-9947-07-04194-3
PII: S 0002-9947(07)04194-3
Keywords: Quadratic conditional variances, harnesses, orthogonal martingale polynomials, hypergeometric orthogonal polynomials
Received by editor(s): June 8, 2005
Received by editor(s) in revised form: September 26, 2005
Posted: June 13, 2007
Additional Notes: This research was partially supported by NSF grants \#INT-0332062, \#DMS-0504198, and by the C.P. Taft Memorial Fund.
Copyright of article: Copyright 2007, American Mathematical Society

			Journals Home Search Author Info Subscribe Tech Support Help
ISSN 1088-6850(e) ISSN 0002-9947(p)

Previous issue \| Table of contents \| Next issue Articles in press \| Recently posted articles \| Most recent issue \| All issues Previous Article \| Next Article


	Comments: webmaster@ams.org © Copyright 2009, American Mathematical Society Privacy Statement		Search the AMS