Hostname: page-component-8448b6f56d-sxzjt Total loading time: 0 Render date: 2024-04-18T23:00:46.363Z Has data issue: false hasContentIssue false
  • English
  • Français

Stochastic LU factorizations, Darboux transformations and urn models

Part of: Markov processes Nontrigonometric harmonic analysis

Published online by Cambridge University Press:  16 November 2018

F. Alberto Grünbaum  and
Manuel D. de la Iglesia
F. Alberto Grünbaum*
Affiliation:
University of California, Berkeley
Manuel D. de la Iglesia*
Affiliation:
Universidad Nacional Autónoma de México
*
* Postal address: Department of Mathematics, University of California, Berkeley, CA 94720, USA.
** Postal address: Instituto de Matemáticas, Universidad Nacional Autónoma de México, Circuito Exterior, C.U., 04510, Ciudad de México, México. Email address: mdi29@im.unam.mx
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider upper‒lower (UL) (and lower‒upper (LU)) factorizations of the one-step transition probability matrix of a random walk with the state space of nonnegative integers, with the condition that both upper and lower triangular matrices in the factorization are also stochastic matrices. We provide conditions on the free parameter of the UL factorization in terms of certain continued fractions such that this stochastic factorization is possible. By inverting the order of the factors (also known as a Darboux transformation) we obtain a new family of random walks where it is possible to state the spectral measures in terms of a Geronimus transformation. We repeat this for the LU factorization but without a free parameter. Finally, we apply our results in two examples; the random walk with constant transition probabilities, and the random walk generated by the Jacobi orthogonal polynomials. In both situations we obtain urn models associated with all the random walks in question.

Keywords

Darboux transformationLU factorizationorthogonal polynomialrandom walkurn model

MSC classification

Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Secondary: 33C45: Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) 42C05: Orthogonal functions and polynomials, general theory
Type
Research Papers
Information
Journal of Applied Probability , Volume 55 , Issue 3 , September 2018 , pp. 862 - 886
Copyright
Copyright © Applied Probability Trust 2018 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Castro, M. M. and Grünbaum, F. A. (2013). On a seminal paper by Karlin and McGregor. SIGMA 9, 020.Google Scholar
[2]Chihara, T. S. (1978). An Introduction to Orthogonal Polynomials. Gordon and Breach, New York.Google Scholar
[3]Dette, H., Reuther, B., Studden, W. J. and Zygmunt, M. (2006). Matrix measures and random walks with a block tridiagonal transition matrix. SIAM J. Matrix Anal. Appl. 29, 117–142.Google Scholar
[4]Feller, W. (1968). An Introduction to Probability Theory and Its Applications, Vol. 1, 3rd edn. John Wiley, New York.Google Scholar
[5]Grassmann, W. K. (1987). Means and variances of time averages in Markovian environments. Europ. J. Operat. Res. 31, 132–139.Google Scholar
[6]Grassmann, W. K. (1993). Means and variances in Markov reward systems. In Linear Algebra, Markov Chains and Queueing Models, Springer, New York, pp. 193–204.Google Scholar
[7]Grünbaum, F. A. (2008). Random walks and orthogonal polynomials: some challenges. In Probability, Geometry and Integrable Systems (Math. Sci. Res. Inst. Publ. 55), Cambridge University Press, pp. 241–260.Google Scholar
[8]Grünbaum, F. A. (2010). An urn model associated with Jacobi polynomials. Comm. App. Math. Comput. Sci. 5, 55–63.Google Scholar
[9]Grünbaum, F. A. (2011). The Darboux process and a noncommutative bispectral problem: some explorations and challenges. In Geometric Aspects of Analysis and Mechanics (Progr. Math. 292), Birkhäuser, New York, pp. 161–177.Google Scholar
[10]Grünbaum, F. A. and Haine, L. (1996). Orthogonal polynomials satisfying differential equations: the role of the Darboux transformation. In Symmetries and Integrability of Difference Equations, American Mathematical Society, Providence, RI, pp. 143–154.Google Scholar
[11]Grünbaum, F. A., Haine, L. and Horozov, E. (1999). Some functions that generalize the Krall-Laguerre polynomials. J. Comput. Appl. Math. 106, 271–297.Google Scholar
[12]Grünbaum, F. A., Pacharoni, I. and Tirao, J. (2013). Two stochastic models of a random walk in the U(n)-spherical duals of U(n+1). Ann. Mat. Pura Appl. (4) 192, 447–473.Google Scholar
[13]Heyman, D. P. (1995). A decomposition theorem for infinite stochastic matrices. J. Appl. Prob. 32, 893–901.Google Scholar
[14]Karlin, S. and McGregor, J. L. (1957). The differential equations of birth-and-death processes, and the Stieltjes moment problem. Trans. Amer. Math. Soc. 85, 489–546.Google Scholar
[15]Karlin, S. and McGregor, J. (1957). The classification of birth and death processes. Trans. Amer. Math. Soc. 86, 366–400.Google Scholar
[16]Karlin, S. and McGregor, J. (1959). Random walks. IIlinois J. Math. 3, 66–81.Google Scholar
[17]Matveev, V. B. and Salle, M. A. (1979). Differential-difference evolution equations II. Darboux transformation for the Toda lattice. Lett. Math. Phys. 3, 425–429.Google Scholar
[18]Spiridonov, V. and Zhedanov, A. (1999). Self-similarity, spectral transformations and orthogonal and biorthogonal polynomials. In Self-Similar Systems, Joint Institute of Nuclear Research, Dubna, pp. 349–361.Google Scholar
[19]Vigon, V. (2013). LU-factorization versus Wiener-Hopf factorization for Markov chains. Acta Appl. Math. 128, 1–37.Google Scholar
[20]Wall, H. S. (1948). Analytic Theory of Continued Fractions. Van Nostrand, New York.Google Scholar
[21]Yoon, G. J. (2002). Darboux transforms and orthogonal polynomials. Bull. Korean Math. Soc. 39, 359–376.Google Scholar
[22]Zhedanov, A. (1997). Rational spectral transformations and orthogonal polynomials. J. Comput. Appl. Math. 85, 67–86.Google Scholar