Related papers: Convergence of a Second Order Markov Chain

The notion of a successful coupling of Markov processes, based on the idea that both components of the coupled system ``intersect'' in finite time with probability one, is extended to cover situations when the coupling is unnecessarily…

Second order recurrence of a $d$-dimensional diffusion with an additive Wiener process, with switching, and with one recurrent and one transient regime and constant switching intensities is established under suitable conditions. The…

We obtain a perfect sampling characterization of weak ergodicity for backward products of finite stochastic matrices, and equivalently, simultaneous tail triviality of the corresponding nonhomogeneous Markov chains. Applying these ideas to…

We study trace theoretic concurrent systems. This setting encompasses safe (1-bounded) Petri nets. We introduce a notion of irreducible concurrent system and we prove the equivalence between irreducibility and a "spectral property". The…

Consider a Markov process $\{\Phi(t) : t\geq 0\}$ evolving on a Polish space ${\sf X}$. A version of the $f$-Norm Ergodic Theorem is obtained: Suppose that the process is $\psi$-irreducible and aperiodic. For a given function $f\colon{\sf…

We study a special case of the vertex splitting model which is a recent model of randomly growing trees. For any finite maximum vertex degree $D$, we find a one parameter model, with parameter $\alpha \in [0,1]$ which has a so--called…

We consider a discrete-time system of n coupled random vectors, a.k.a. interacting particles. The dynamics involve a vanishing step size, some random centered perturbations, and a mean vector field which induces the coupling between the…

Upon the introduction of the Metropolis algorithm, the question of how many steps in the Markov chain were needed to achieve convergence to stationarity became apparent. The convergence was rather slow, i.e. for a process on $n$ states the…

Random spanning trees are among the most prominent determinantal point processes. We give four examples of random spanning trees on ladder-like graphs whose rungs form stationary renewal processes or regenerative processes of order two,…

Fix a finite dimension $d \geq 2$ and a fixed rank-1 PVM $M=\{|e_1\rangle\langle e_1|,\ldots,|e_d\rangle\langle e_d|\}$ on ${\bf C}^d$. Let $P_M:\mathbb{CP}^{d-1}\to\Delta^{d-1}$ be a readout map on pure states. We prove that three…

In many iterative optimization methods, fixed-point theory enables the analysis of the convergence rate via the contraction factor associated with the linear approximation of the fixed-point operator. While this factor characterizes the…

We present an efficient finite difference method for the approximation of second derivatives, with respect to system parameters, of expectations for a class of discrete stochastic chemical reaction networks. The method uses a coupling of…

The transition matrix of a Markov chain $(X_k,k\geq 0)$ on a finite or infinite rooted tree is said to be almost upper-directed if, given $X_k$, the node $X_{k+1}$ is either a descendant of $X_k$ or the parent of $X_k$. It is said to be…

This work presents a low-rank tensor model for multi-dimensional Markov chains. A common approach to simplify the dynamical behavior of a Markov chain is to impose low-rankness on the transition probability matrix. Inspired by the success…

We use the $f-divergence$ also called relative entropy as a measure of diversity between probability densities and review its basic properties. In the sequence we define a few objects which capture relevant information from the sample of a…

Nonlinear Markov chains with finite state space have been introduced in Kolokoltsov (2010). The characteristic property of these processes is that the transition probabilities do not only depend on the state, but also on the distribution of…

Using elementary methods, we prove that for a countable Markov chain $P$ of ergodic degree $d > 0$ the rate of convergence towards the stationary distribution is subgeometric of order $n^{-d}$, provided the initial distribution satisfies…

Monoid actions of trace monoids over finite sets are powerful models of concurrent systems---for instance they encompass the class of 1-safe Petri nets. We characterise Markov measures attached to concurrent systems by finitely many…

Markov chain Monte Carlo is an inherently serial algorithm. Although likelihood calculations for individual steps can sometimes be parallelized, the serial evolution of the process is widely viewed as incompatible with parallelization,…

We suggest an approach to obtaining general two-sided bounds on the rate of convergence in terms of special "weighted" norms related to total variation. Some important classes of continuous-time Markov chains are considered:…