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This review paper, written for the second edition of the Handbook of Markov Chain Monte Carlo, provides an introduction to the study of convergence analysis for Markov chain Monte Carlo (MCMC), aimed at researchers new to the field. We…

Misperceptions about extreme dependencies between different financial assets have been an im- portant element of the recent financial crisis. This paper studies inhomogeneity in dependence structures using Markov switching regular vine…

We establish convergence to an invariant measure as time tends to infinity, for a large class of (possibly non-Markovian) stochastic volatility models. Our arguments are based on a novel coupling idea for Markov chains which also extends to…

In many systems consisting of interacting subsystems, the complex interactions between elements can be represented using multilayer networks. However percolation, key to understanding connectivity and robustness, is not trivially…

We develop a theory of weak Poincar\'e inequalities to characterize convergence rates of ergodic Markov chains. Motivated by the application of Markov chains in the context of algorithms, we develop a relevant set of tools which enable the…

The concepts of probability, statistics and stochastic theory are being successfully used in structural engineering. Markov Chain modelling is a simple stochastic process model that has found its application in both describing stochastic…

There is a well-established theory linking certain semi-Markov chains and continuous-time random walks to time-fractional equations and anomalous diffusion. In this work, we go beyond the semi-Markov framework by considering some…

Building upon earlier work in which axioms were formulated for multivariate measures of concordance, we examine properties of such measures. In particular, we examine the relations between the measure of concordance of an $n$-copula and the…

We survey information-theoretic approaches to the reduction of Markov chains. Our survey is structured in two parts: The first part considers Markov chain coarse graining, which focuses on projecting the Markov chain to a process on a…

The aim of this work is to give an introduction to the theoretical background and computational complexity of Markov chain Monte Carlo methods. Most of the mathematical results related to the convergence are not found in most of the…

In this review-type paper written at the occasion of the Oberwolfach workshop {\em One-sided vs. Two-sided stochastic processes} (february 22-29, 2020), we discuss and compare Markov properties and generalisations thereof in more…

Looking at bivariate copulas from the perspective of conditional distributions and considering weak convergence of almost all conditional distributions yields the notion of weak conditional convergence. At first glance, this notion of…

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…

We consider a degenerate stochastic differential equation that has a sticky point in the Markov process sense. We prove that weak existence and weak uniqueness hold, but that pathwise uniqueness does not hold nor does a strong solution…

We introduce a new functional measure of tail dependence for weakly dependent (asymptotically independent) random vectors, termed weak tail dependence function. The new measure is defined at the level of copulas and we compute it for…

In this paper, we mainly consider about the existence and uniqueness of global weak solutions for the two-component Novikov system. We first recall some results and definitions of strong solutions and weak solutions for the system, then by…

We study the large deviations of Markov chains under the sole assumption that the state space is discrete. In particular, we do not require any of the usual irreducibility and exponential tightness assumptions. Using subadditive arguments,…

We study the Markovianity of a composite system and its subsystems. We show how the dissipative nature of a subsystem's dynamics can be modified without having to change properties of the composite system environment. By preparing different…

For many Markov chains of practical interest, the invariant distribution is extremely sensitive to perturbations of some entries of the transition matrix, but insensitive to others; we give an example of such a chain, motivated by a problem…

The basic question in perturbation analysis of Markov chains is: how do small changes in the transition kernels of Markov chains translate to chains in their stationary distributions? Many papers on the subject have shown, roughly, that the…