Article
Physics, Multidisciplinary
Abdessatar Souissi, El Gheteb Soueidy, Abdessatar Barhoumi
Summary: We study the psi-mixing property for entangled Markov chains and prove that those satisfying the Markov-Dobrushin condition are psi-mixing. Moreover, the restriction of the underlying QMC to the diagonal algebra results in a classical mixing Markov chains with the same exponential rate of convergence.
PHYSICA A-STATISTICAL MECHANICS AND ITS APPLICATIONS
(2023)
Article
Mathematics, Applied
Nicos Georgiou, Enrico Scalas
Summary: This article discusses the changes that occur in the mixing time of a Markov chain when the integer step index n is replaced with a random counting process N(t). The focus is on the case where N(t) follows a power-law distribution with index beta. Specifically, the article studies the situation when beta is an element of (0, 1), leading to infinite expectation for the inter-arrival times, and further investigates the scenario where the inter-arrival times follow the Mittag-Leffler distribution of order beta.
FRACTIONAL CALCULUS AND APPLIED ANALYSIS
(2022)
Article
Computer Science, Software Engineering
Prateek Bhakta, Sarah Miracle, Dana Randall, Amanda Pascoe Streib
Summary: This article investigates the mixing time of a Markov chain on biased permutations and suggests that the chain mixes rapidly when the probabilities are positively biased.
RANDOM STRUCTURES & ALGORITHMS
(2022)
Article
Mathematics
Ajitha K. B. Shenoy, Smitha N. Pai
Summary: This study investigates the relationship between the magnification of a search graph and the success of local search-based metaheuristic algorithms. It shows that the ergodic reversible Markov chain induced by these algorithms is inversely proportional to magnification. Therefore, it is beneficial to use a search space with large magnification for optimization problems. The study also demonstrates that the Metropolis Algorithm can mix rapidly if the search graph has a large magnification. The results are applied to the well-studied 0/1-Knapsack Problem, showing that Markov Chains associated with local search-based metaheuristics mix rapidly.
Article
Mathematics
Andreas C. Georgiou, Alexandra Papadopoulou, Pavlos Kolias, Haris Palikrousis, Evanthia Farmakioti
Summary: Semi-Markov processes generalize the Markov chains framework by utilizing abstract sojourn time distributions for enhanced accuracy in modeling stochastic phenomena. This paper aims to provide closed analytic forms for three types of probabilities describing attributes of considerable research interest in semi-Markov modeling, with the development of non-homogeneous in time recursive relations and corresponding geometric transforms. The theoretical results are illustrated using data from human DNA sequences.
Article
Mathematics
Lixing Han, Kelun Wang, Jianhong Xu
Summary: This study explores the relationship between ergodicity and irreducibility of transition probability tensors, introducing the concepts of first passage times and mean first passage times in tensor form for higher order Markov chains, and proving various properties of these quantities. Numerical examples are provided to illustrate the results.
LINEAR & MULTILINEAR ALGEBRA
(2022)
Article
Mathematics, Applied
Yves F. Atchade
Summary: This paper introduces a concept of approximate spectral gap to analyze the mixing time of reversible Markov chain Monte Carlo algorithms. The study focuses on cases where the usual spectral gap is degenerate or almost degenerate. The results show that properly tuned algorithms have a mixing time that grows at most polynomially with dimension, and improve when the posterior distribution contracts towards the true model with a well chosen initial distribution.
SIAM JOURNAL ON MATHEMATICS OF DATA SCIENCE
(2021)
Article
Mathematical & Computational Biology
David F. Anderson, Jinsu Kim
Summary: The past few decades have witnessed extensive research on the existence, form, and properties of stationary distributions in stochastically modeled reaction networks. The rate of convergence of the process distribution to the stationary distribution has not been widely studied in the reaction network literature, except for models with a state space of non-negative integers. This paper addresses this research gap by characterizing the rate of convergence for two classes of stochastically modeled reaction networks and establishing exponential ergodicity using a Foster-Lyapunov criteria. The paper also demonstrates uniform convergence over the initial state for one of the classes.
MATHEMATICAL BIOSCIENCES AND ENGINEERING
(2023)
Article
Mathematics
Robert M. Anderson, Haosui Duanmu, Aaron Smith
Summary: Peres and Sousi demonstrated that the mixing times of reversible Markov chains on finite state spaces are equivalent up to a universal constant. By utilizing nonstandard analysis, this result was extended to reversible Markov chains on compact state spaces with the strong Feller property.
CANADIAN MATHEMATICAL BULLETIN-BULLETIN CANADIEN DE MATHEMATIQUES
(2021)
Article
Mathematics, Applied
Omer Angel, Yinon Spinka
Summary: In this study, it is demonstrated that the stationary version of an ergodic Markov chain with exponential tails in return times is a finitary factor of an independent and identically distributed (i.i.d.) process. The essential step involves proving that a stationary renewal process with exponential tails in the jump distribution, not supported on a proper subgroup of Z, is also a finitary factor of an i.i.d. process.
ERGODIC THEORY AND DYNAMICAL SYSTEMS
(2021)
Article
Statistics & Probability
Rafael Chiclana, Yuval Peres
Summary: This article investigates random walks on finite spherically symmetric trees and obtains a result that the ratio of relaxation time and mixing time is bounded. This implies the absence of pre-cutoff phenomenon regardless of the size of the tree, which also applies to continuous time random walks. The hitting times of vertices are also examined. The results show that the stability of the problem under rough isometries.
ELECTRONIC COMMUNICATIONS IN PROBABILITY
(2022)
Article
Statistics & Probability
Keunwoo Lim
Summary: This study investigates the cutoff phenomena of certain cyclic dynamics on the hypercube, showing that these dynamics, represented by irreversible Markov chains, exhibit cutoff phenomena.
MARKOV PROCESSES AND RELATED FIELDS
(2022)
Article
Computer Science, Software Engineering
Emilio De Santis
Summary: In this paper, it is shown that for any given digraph G, a 1-dependent Markov chain can be constructed along with n hitting times to address paradoxes in probability theory, particularly related to non-transitive dice.
RANDOM STRUCTURES & ALGORITHMS
(2021)
Review
Environmental Sciences
Sadjia Hamdad, Mourad Lazri, Yacine Mohia, Karim Labadi, Soltane Ameur
Summary: The paper presents a spatiotemporal analysis of the dry/wet phenomenon in the rainy period in northern Algeria for predicting drought. Artificial neural networks (ANN) and Markov chains (MC) were used to analyze the behavior of this phenomenon. The predictions from both methods showed good agreement with actual satellite data in short-term evaluations. Long-term predictions were also made and compared with simulations from CMIP6, indicating consistent trends among the three models (ANN, MC, and CMIP6).
JOURNAL OF THE INDIAN SOCIETY OF REMOTE SENSING
(2023)
Article
Mathematics
Lixing Han, Jianhong Xu
Summary: Within a tensor framework, this paper studies the relations between the ergodicity, ever-reaching probabilities, and mean first passage times of higher order Markov chains. It also proposes block iterative and block direct algorithms for computing these quantities.
LINEAR & MULTILINEAR ALGEBRA
(2022)
