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Conditional moments of q-Meixner processes
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  • Published: 12 September 2004

Conditional moments of q-Meixner processes

  • Włodzimierz Bryc1 &
  • Jacek Wesołowski2 

Probability Theory and Related Fields volume 131, pages 415–441 (2005)Cite this article

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  • 41 Citations

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Abstract.

We show that stochastic processes with linear conditional expectations and quadratic conditional variances are Markov, and their transition probabilities are related to a three-parameter family of orthogonal polynomials which generalize the Meixner polynomials. Special cases of these processes are known to arise from the non-commutative generalizations of the Lévy processes.

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Author information

Authors and Affiliations

  1. Department of Mathematics, University of Cincinnati, 210025, Cincinnati, OH, 45221–0025, USA

    Włodzimierz Bryc

  2. Faculty of Mathematics and Information Science, Warsaw University of Technology, pl. Politechniki 1, 00-661, Warszawa, Poland

    Jacek Wesołowski

Authors
  1. Włodzimierz Bryc
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  2. Jacek Wesołowski
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Corresponding author

Correspondence to Włodzimierz Bryc.

Additional information

Mathematics Subject Classification (2000): 60J25

Research partially supported by NSF grant #INT-0332062, by the C.P. Taft Memorial Fund, and University of Cincinnati’s Summer Faculty Research Fellowship Program

AcknowledgementPart of the research of WB was conducted while visiting the Faculty of Mathematics and Information Science of Warsaw University of Technology. The authors thank M. Bożejko for bringing to their attention several references, to Hiroaki Yoshida for information pertinent to Theorem 4.3, and to M. Anshelevich, W. Matysiak, R. Speicher, P. Szabłowski, and M. Yor for helpful comments and discussions. Referee’s comments lead to several improvements in the paper.

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Bryc, W., Wesołowski, J. Conditional moments of q-Meixner processes. Probab. Theory Relat. Fields 131, 415–441 (2005). https://doi.org/10.1007/s00440-004-0379-2

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  • Received: 18 February 2004

  • Revised: 12 May 2004

  • Published: 12 September 2004

  • Issue Date: March 2005

  • DOI: https://doi.org/10.1007/s00440-004-0379-2

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Key words or phrases:

  • Quadratic conditional variances
  • Harnesses
  • Polynomial martingales
  • Hypergeometric orthogonal polynomials
  • Free Lévy processes
  • Classical versions of non-commutative processes
  • 2-Meixner processes
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