Related papers: The spectral matrices associated with the stochast…
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…
We consider stochastic UL and LU block factorizations of the one-step transition probability matrix for a discrete-time quasi-birth-and-death process, namely a stochastic block tridiagonal matrix. The simpler case of random walks with only…
We consider a new way of factorizing the transition probability matrix of a discrete-time birth-death chain on the integers by means of an absorbing and a reflecting birth-death chain to the state 0 and viceversa. First we will consider…
We give a probabilistic interpretation of the associated Jacobi polynomials, which can be constructed from the three-term recurrence relation for the classical Jacobi polynomials by shifting the integer index $n$ by a real number $t$. Under…
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 discrete-time quasi-birth-death process on the…
Recently a spectral Favard theorem for bounded banded lower Hessenberg matrices that admit a positive bidiagonal factorization was presented. These type of matrices are oscillatory. In this paper the Lima-Loureiro hypergeometric multiple…
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 its transpose, which are nonnegative…
We study LU-type factorizations of the infinitesimal generator of a birth--death process on $\mathbb{N}_0$. Our goal is to characterize those factorizations whose Darboux transformations (that is, inverting the order of the factors) yield…
The random walk to be considered takes place in the d- spherical dual of the group U(n + 1), for a fixed finite dimensional irreducible representation d of U(n). The transition matrix comes from the three term recursion relation satisfied…
We investigate the eigenvalue statistics of random Bernoulli matrices, where the matrix elements are chosen independently from a binary set with equal probability. This is achieved by initiating a discrete random walk process over the space…
A general theorem on factorization of matrices with polynomial entries is proven and it is used to reduce polynomial Darboux matrices to linear ones. Some new examples of linear Darboux matrices are discussed.
In this work, we give some criteria that allow us to decide when two sequences of matrix-valued orthogonal polynomials are related via a Darboux transformation and to build explicitly such transformation. In particular, they allow us to see…
Darboux transformation of a second-order linear differential operator is a well-known technique with many applications in mathematics and physics. We study Darboux transformation from the point of view of Markov semigroups of diffusion…
In this paper, we study complex Jacobi matrices obtained by the Christoffel and Geronimus transformations at a nonreal complex number, including the properties of the corresponding sequences of orthogonal polynomials. We also present some…
Let J be a monic Jacobi matrix associated with the Cauchy transform F of a probability measure. We construct a pair of the lower and upper triangular block matrices L and U such that J=LU and the matrix J_c=UL is a monic generalized Jacobi…
The foundational work of Karlin and McGregor established a powerful connection between random walks with tridiagonal transition matrices and the theory of orthogonal polynomials. We consider a particular extension of this framework, where…
Given a non-negative Jacobi matrix describing higher order recurrence relations for multiple orthogonal polynomials of type~II and corresponding linear forms of type I, a general strategy for constructing a pair of stochastic matrices, dual…
We consider random walk on a finite group $G$ as follows. We can consider $G$ as a group of substitutions. Randomly (i.e. with probability $U(g)=|G|^{-1}$ ) we choose a substitution $g \in G$ and execute it twice in a row, i.e. execute a…
Spherically symmetric random walks in arbitrary dimension $D$ can be described in terms of Gegenbauer (ultraspherical) polynomials. For example, Legendre polynomials can be used to represent the special case of two-dimensional spherically…
Let d\mu(t) be a probability measure on [0,+\infty) such that its moments are finite. Then the Cauchy-Stieltjes transform S of d\mu(t) is a Stieltjes function, which admits an expansion into a Stieltjes continued fraction. In the present…