Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2509.01761

Matrix-Valued Orthogonal Polynomials Related to a Class of Random Walk

P. Roman·S. Menchon·Y. Yin
View original source ↗Read source document ↗Export metadata ↓

Abstract

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 the transition matrix is given by a polynomial in a tridiagonal matrix. This generalization leads to transitions beyond the nearest neighbors. We investigate the matrix-valued orthogonal polynomials associated with these extended models, derive the corresponding matrix-valued measure of orthogonality explicitly, and analyze how spectral properties of the transition matrix relate to probabilistic features of the random walk. As an application, we study generalized Ehrenfest models that incorporates longer-range transitions.

Explore related subjects

math.PR

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

P. Roman, S. Menchon, Y. Yin. 2025-09-01. Matrix-Valued Orthogonal Polynomials Related to a Class of Random Walk. https://arxiv.org/abs/2509.01761

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

The extremal process of a cascading family of branching Brownian motion

We study the asymptotic behaviour of the extremal process of a cascading family of branching Brownian motions. This is a particle system on the real line such that each particle has a type in addition to his position. Particles of type $1$ move on the real line according to Brownian motions and branch at rate $1$ into two children of type $1$. Furthermore, at rate $α$, they give birth to children too of type $2$. Particles of type $2$ move according to standard Brownian motion and branch at rate $1$, but cannot give birth to descendants of type $1$. We obtain the asymptotic behaviour of the extremal process of particles of type $2$.

math.PR↗
arXiv2026

On the Wasserstein distance between a hyperuniform point process and its mean

We study the existence of bounds on the expected $p$-Wasserstein distance between a random measure and its mean under the assumption that the $p$-th centered moments of the counting statistics are controlled uniformly in space. The average Wasserstein transport cost is shown to be bounded from above and from below by some multiples of the number of points. $D$-dimensional versions of those results are also obtained. As a corollary, we prove that for any value of $p\geq 1$ the Ginibre point process can be seen as a perturbed lattice with identically distributed perturbations with a finite $p$-th moment.

math.PR↗
arXiv2026

Breuer-Major Theorems for Hilbert Space-Valued Random Variables

Let $\{X_k\}_{k\in\mathbb Z}$ be a stationary Gaussian process with values in a separable Hilbert space $\mathcal H_1$, and let $G:\mathcal H_1\to\mathcal H_2$ be a measurable map into another separable Hilbert space $\mathcal H_2$. We derive a central limit theorem for the centered normalized partial sums of the Hilbert space-valued subordinated process $\{G[X_k]\}_{k\in\mathbb Z}$. Our result holds under either of two sets of sufficient conditions, formulated in terms of the transformation $G$ and the temporal and cross-sectional dependence structure of $\{X_k\}_{k\in\mathbb Z}$. These conditions coincide in finite dimensions but lead to genuinely different phenomena in the infinite-dimensional setting. The proof relies on the recently developed Fourth Moment Theorem on Hilbert spaces, leveraging tools from the infinite-dimensional Malliavin-Stein framework. We also provide continuous-time and quantitative versions of the central limit theorem. In a series of examples, we recover and strengthen limit theorems for a wide array of statistics relevant in functional data analysis, and present, as an application of our result, a novel limit theorem in the framework of neural operators.

math.PR↗