arXiv subjects

Wojciech Matysiak

Publications and source records attributed to Wojciech Matysiak.

11 recordsLinked to original sources

arXiv2023

The asymptotic topology of the multineighbor complex of a random graph

We introduce the multineighbor complex of a graph, which is a simplicial complex in which a simplex is a subset of the graph with a sufficient number of mutual neighbors. We investigate the asymptotic homological properties of such complexes for the Erdos-Renyi random graphs and obtain a number of vanishing and nonvanishing results. We use this construction to perform a topological data analysis classification of noisy synthetic point clouds obtaining favorable accuracy as obtained by the standard methods. The case when there is a single neighbor has been studied earlier by Mathew Kahle.

math.CO↗

arXiv2019

Multiplicity free actions, birth and death processes on partitions, and Biane's quantum Ornstein-Uhlenbeck semigroups

We introduce a family of multivariate continuous-time pure birth and pure death chains, with birth and death rates defined in terms of the generalized binomial coefficients for multiplicity free actions. The state spaces for some of the introduced processes are some sets of partitions (equivalently, Young diagrams). The chains turn out to be the classical Markov processes obtained by restricting Biane's quantum Ornstein-Uhlenbeck semigroups to commutative C*-algebras related to Gelfand pairs built on Heisenberg groups.

math.PR↗

arXiv2016

Hypergroups and Quantum Bessel Processes of Non-integer Dimensions

It is demonstrated how to use certain family of commutative hypergroups to provide a universal construction of Biane's quantum Bessel processes of all dimensions not smaller than 1. The classical Bessel processes BES$(δ)$ are analogously constructed with the aid of the Bessel-Kingman hypergroups for all, not necessarily integer, dimensions $δ\ge1$.

math.PR↗

arXiv2011

Moments and $q$-commutators of noncommutative random vectors

A method for computing the mixed moments of (not necessarily commutative) random vectors from the first order moments, the $q$-commutators between the annihilation and creation operators, and the $q$-commutators between the annihilation and preservation operators, is presented. The method is illustrated by a relevant characterization of $q$-Gaussian vectors.

math.PR↗

arXiv2011

Wilson's 6-j laws and stitched Markov processes

We show how to insert time into the parameters of the Wilson's 6-j laws to construct discrete Markov chains with these laws. By a quadratic transformation we convert them into Markov processes with linear regressions and quadratic conditional variances. Further conversion into the "standard form" gives "quadratic harnesses" with "classical" value of parameter gamma. A random-parameter-representation of the original Markov chain allows us to stitch together two copies of the process, extending time domain of the quadratic harness from (0,1) to all t>0.

math.PR↗

arXiv2010

Free Quadratic Harness

Free quadratic harness is a Markov process from the class of quadratic harnesses, i.e. processes with linear regressions and quadratic conditional variances. The process has recently been constructed for a restricted range of parameters in the paper "Askey-Wilson polynomials, quadratic harnesses and martingales" by W. Bryc and J. Wesołowski using Askey--Wilson polynomials. Here we provide a self-contained construction of the free quadratic harness for all values of parameters.

math.PR↗

arXiv2008

The bi-Poisson process: a quadratic harness

This paper is a continuation of our previous research on quadratic harnesses, that is, processes with linear regressions and quadratic conditional variances. Our main result is a construction of a Markov process from given orthogonal and martingale polynomials. The construction uses a two-parameter extension of the Al-Salam--Chihara polynomials and a relation between these polynomials for different values of parameters.

math.PR↗

arXiv2006

Generalized stationary random fields with linear regressions - an operator approach

Existence, $L^2$-stationarity and linearity of conditional expectations $\wwo{X_k}{...,X_{k-2},X_{k-1}}$ of square integrable random sequences $\mathbf{X}=(X_{k})_{k\in\mathbb{Z}}$ satisfying \[ \wwo{X_k}{...,X_{k-2},X_{k-1},X_{k+1},X_{k+2},...}=\sum_{j=1}^\infty b_j(X_{k-j}+X_{k+j}) \] for a real sequence $(b_n)_{n\in\nat}$, is examined. The analysis is reliant upon the use of Laurent and Toeplitz operator techniques.

math.PR↗

arXiv2005

Bryc's random fields: the existence and distributions analysis

We examine problem of existence of stationary random fields with linear regressions and quadratic conditional variances, introduced by Bryc in "Stationary random fields with linear regressions" (Annals of Probability 29, No. 1, 504-519). Distributions of the fields are identified and almost complete description of the possible sets of parameters defining the first two conditional moments is given. This note almost solves Bryc's problem concerning fields undetermined by moments - the only remaining set of parameters for which the existence of Bryc's fields is unclear has Lebesgue measure zero.

math.PR↗

arXiv2005

Quadratic Harnesses, q-commutations, and orthogonal martingale polynomials

We introduce the quadratic harness condition and show that integrable quadratic harnesses have orthogonal martingale polynomials with a three step recurrence that satisfies a q-commutation relation. This implies that quadratic harnesses are essentially determined uniquely by five numerical constants. Explicit recurrences for the orthogonal martingale polynomials are derived in several cases of interest.

math.PR↗

arXiv2003

Probabilistic aspects of Al-Salam-Chihara polynomials

We solve the connection coefficient problem between the Al-Salam-Chihara polynomials and the q-Hermite polynomials, and we use the resulting identity to answer a question from probability theory. We also derive the distribution of some Al-Salam-Chihara polynomials, and compute determinants of related Hankel matrices.

math.CA↗