Stochastic LU factorizations, Darboux transformations and urn models

We consider UL (and LU) decompositions of the one-step transition probability matrix of a random walk with state space the nonnegative integers, with the condition that both upper and lower triangular matrices in the factorization are also stochastic matrices. We give conditions on the free parameter of the UL factorization in terms of certain continued fraction such that this stochastic factorization is possible. By inverting the order of multiplication (also known as a Darboux transformation) we get a new family of random walks where it is possible the identify the spectral measures in terms of a Geronimus transformation. The same can be done for the LU factorization but now 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 give urn models associated with all the random walks in question.

[1]  Samuel Karlin,et al.  The classification of birth and death processes , 1957 .

[2]  H. Wall,et al.  Analytic Theory of Continued Fractions , 2000 .

[3]  T. Chihara,et al.  An Introduction to Orthogonal Polynomials , 1979 .

[4]  Vincent Vigon.  LU-Factorization Versus Wiener-Hopf Factorization for Markov Chains , 2013 .

[5]  D. Heyman.  A decomposition theorem for infinite stochastic matrices , 1995 .

[6]  Random walks and orthogonal polynomials: some challenges , 2007, math/0703375.

[7]  F. Alberto Grünbaum.  The Darboux process and a noncommutative bispectral problem: some explorations and challenges , 2008 .

[8]  Holger Dette,et al.  Matrix Measures and Random Walks with a Block Tridiagonal Transition Matrix , 2006, SIAM J. Matrix Anal. Appl..

[9]  F. Alberto Grünbaum,et al.  Some functions that generalize the Krall-Laguerre polynomials , 1999 .

[10]  F. A. Grünbaum,et al.  Orthogonal polynomials satisfying differential equations: The role of the Darboux transformation , 1996 .

[11]  Winfried K. Grassmann.  Means and Variances in Markov Reward Systems , 1993 .

[12]  William Feller,et al.  An Introduction to Probability Theory and Its Applications , 1967 .

[13]  Alexei Zhedanov,et al.  Rational spectral transformations and orthogonal polynomials , 1997 .

[14]  V. Matveev,et al.  Differential-difference evolution equations. II (Darboux transformation for the Toda lattice) , 1979 .

[15]  Gangjoon Yoon,et al.  DARBOUX TRANSFORMS AND ORTHOGONAL POLYNOMIALS , 2002 .

[16]  F. Grünbaum,et al.  Two stochastic models of a random walk in the U(n)-spherical duals of U(n + 1) , 2010, 1010.0720.

[17]  F. A. Grüunbaum.  Random walks and orthogonal polynomials: some challenges , 2008 .

[18]  S. Karlin,et al.  The differential equations of birth-and-death processes, and the Stieltjes moment problem , 1957 .

[19]  Sariel Har-Peled,et al.  Random Walks , 2021, Encyclopedia of Social Network Analysis and Mining.

[20]  F. Grünbaum.  An urn model associated with Jacobi polynomials , 2009 .

[21]  Winfried K. Grassmann,et al.  Means and variances of time averages in Markovian environments , 1987 .

[22]  F. Grunbaum,et al.  On a Seminal Paper by Karlin and McGregor , 2013, 1303.0361.