arXiv subjects

Jianhong Xu

Publications and source records attributed to Jianhong Xu.

6 recordsLinked to original sources

arXiv2026

An Alternative Framework for Irreducibility and Primitivity of Nonnegative Tensors

Motivated by some recent studies on higher order Markov chains and well-known characterizations for irreducibility and primitivity of nonnegative matrices, we propose in this paper an alternative framework for irreducibility and primitivity of nonnegative tensors, giving rise to the concepts of s-irreducibility and s-primitivity. This framework includes the relevant results on matrices as its special cases, yet it expands existing results regarding irreducibility and primitivity for tensors. In addition to its tensor theoretic significance, such a framework has important implications for applied fields, especially when it comes to higher order Markov chains.

math.RA↗

arXiv2026

Mean First Passage Times and Fundamental Tensors of Higher Order Markov Chains

The mean first passage times are among the most critical characteristics of a Markov chain. In this paper, we focus on the scenario in which one or more states of a higher order ergodic Markov chain are modified to be absorbing. We prove that the resulting chain has to be absorbing. For a higher order absorbing Markov chain, we prove that the equation its fundamental tensor satisfies must be nonsingular and provide a MATLAB function {\tt fund} for solving the equation. Besides, we connect each horizontal slice of the mean first passage time tensor with a fundamental tensor obtained when one state of a higher order ergodic Markov chain is modified to be absorbing, which also leads to a tensor series representation for selected mean first passage times.

math.PR↗

arXiv2025

Can a Higher Order Markov Chain Be Treated as a First Order Markov Chain?

It is well known that any higher order Markov chain can be associated with a first order Markov chain. In this primarily expository article, we present the first fairly comprehensive analysis of the relationship between higher order and first order Markov chains, together with illustrative examples. Our main objective is to address the central question as posed in the title.

math.PR↗

arXiv2025

HOMC: A MATLAB Package for Higher Order Markov Chains

We present a MATLAB package, which is the first of its kind, for Higher Order Markov Chains (HOMC). It can be used to easily compute all important quantities in our recent works relevant to higher order Markov chains, such as the $k$-step transition tensor, limiting probability distribution, ever-reaching probability tensor, and mean first passage time tensor. It can also be used to check whether a higher order chain is ergodic or regular, to construct the transition matrix of the associated reduced first order chain, and to determine whether a state is recurrent or transient. A key function in the package is an implementation of the tensor ``box'' product which has a probabilistic interpretation and is different from other tensor products in the literature. This HOMC package is useful to researchers and practitioners alike for tasks such as numerical experimentation and algorithm prototyping involving higher order Markov chains.

stat.CO↗

arXiv2025

On Limiting Probability Distributions of Higher Order Markov Chains

The limiting probability distribution is one of the key characteristics of a Markov chain since it shows its long-term behavior. In this paper, for a higher order Markov chain, we establish some properties related to its exact limiting probability distribution, including a sufficient condition for the existence of such a distribution. Our results extend the corresponding conclusions on first order chains. Besides, they complement the existing results concerning higher order chains which rely on approximation schemes or two-phase power iterations. Several illustrative example are also given.

math.PR↗

arXiv2017

On the Convergence of the Accelerated Riccati Iteration Method

In this paper, we establish results fully addressing two open problems proposed recently by I. Ivanov, see Nonlinear Analysis 69 (2008) 4012--4024, with respect to the convergence of the accelerated Riccati iteration method for solving the continuous coupled algebraic Riccati equation, or CCARE for short. These results confirm several desirable features of that method, including the monotonicity and boundedness of the sequences it produces, its capability of determining whether the CCARE has a solution, the extremal solutions it computes under certain circumstances, and its faster convergence than the regular Riccati iteration method.

math.OC↗