Stochastic LU factorizations, Darboux transformations and urn models

  title={Stochastic LU factorizations, Darboux transformations and urn models},
  author={F. Alberto Gr{\"u}nbaum and Manuel Dom{\'i}nguez de la Iglesia},
  journal={J. Appl. Probab.},
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… 
Hypergeometric Multiple Orthogonal Polynomials and Random Walks
The recently found hypergeometric multiple orthogonal polynomials on the step-line by Lima and Loureiro are shown to be random walk polynomials. It is proven that the corresponding Jacobi matrix and
Birth-death chains on a spider: spectral analysis and reflecting-absorbing factorization
We consider discrete-time birth-death chains on a spider, i.e. a graph consisting of N discrete half lines on the plane that are joined at the origin. This process can be identified with a
Oscillatory banded Hessenberg matrices, multiple orthogonal polynomials and random walks
. A spectral Favard theorem for oscillatory bounded banded lower Hessenberg matrices is found. To motivate the relevance of the oscillatory character, the spectral Favard theorem for bounded Jacobi
Quasi-birth-and-death processes and multivariate orthogonal polynomials
Spectral Representation of Birth–Death Processes
  • Mathematics
    Orthogonal Polynomials in the Spectral Analysis of Markov Processes
  • 2021
Spectral Representation of Diffusion Processes
  • Mathematics
    Orthogonal Polynomials in the Spectral Analysis of Markov Processes
  • 2021
  • Orthogonal Polynomials in the Spectral Analysis of Markov Processes
  • 2021


LU-Factorization Versus Wiener-Hopf Factorization for Markov Chains
Our initial motivation was to understand links between Wiener-Hopf factorizations for random walks and LU-factorizations for Markov chains as interpreted by Grassman (Eur. J. Oper. Res.
Two stochastic models of a random walk in the U(n)-spherical duals of U(n + 1)
The random walk to be considered takes place in the δ-spherical dual of the group U(n + 1), for a fixed finite dimensional irreducible representation δ of U(n). The transition matrix comes from the
Matrix Measures and Random Walks with a Block Tridiagonal Transition Matrix
This paper derives sufficient conditions such that the blocks of the n-step block tridiagonal transition matrix of the Markov chain can be represented as integrals with respect to a matrix valued spectral measure.
Some functions that generalize the Krall-Laguerre polynomials
A decomposition theorem for infinite stochastic matrices
  • D. Heyman
  • Mathematics
    Journal of Applied Probability
  • 1995
We prove that every infinite-state stochastic matrix P say, that is irreducible and consists of positive-recurrrent states can be represented in the form I – P=(A – I)(B – S), where A is strictly
Rational spectral transformations and orthogonal polynomials
The Darboux process and a noncommutative bispectral problem: some explorations and challenges
The Darboux process is revisited in a case when scalars are replaced by matrices, i.e., elements of a non-commutative ring, with emphasis on very concrete examples involving 2×2 matrices.
Means and Variances in Markov Reward Systems
In this paper, we study the total reward connected with a Markov reward process from time zero to time m. In particular, we determine the average reward within this time period, as well as its
Random Walks
  • Sariel Har-Peled
  • Mathematics
    Encyclopedia of Social Network Analysis and Mining
  • 2014
4.* Use 3. to show that the average number of visits to a > 0 before returning to the origin is 1 (hint: show that it is closely related to the expectation of some geometric random variable).
We give a new interpretation of Darboux transforms in the context of orthogonal polynomials and find conditions in or-der for any Darboux transform to yield a new set of orthogonal polynomials. We