Related papers: Spectral gap of nonreversible Markov chains
We study the spectral theory of a reversible Markov chain associated to a hypoelliptic random walk on a manifold M. This random walk depends on a parameter h which is roughly the size of each step of the walk. We prove uniform bounds with…
The edge flipping is a non-reversible Markov chain on a given connected graph, which is defined by Chung and Graham in [CG12]. In the same paper, its eigenvalues and stationary distributions for some classes of graphs are identified. We…
By viewing the $N$-simplex as the set of positions of $N-1$ ordered particles on the unit interval, the adjacent walk is the continuous time Markov chain obtained by updating independently at rate 1 the position of each particle with a…
This paper will provide several classes of strictly stationary, countable-state, irreducible, aperiodic Markov chains that are reversible and have finite second moments, such that the central limit theorem fails to hold. The main purpose is…
This paper presents an elementary proof of stochastic stability of a discrete-time reversible Markov chain starting from a Foster-Lyapunov drift condition. Besides its relative simplicity, there are two salient features of the proof: (i) it…
The problem of efficiently sampling from a set of (undirected, or directed) graphs with a given degree sequence has many applications. One approach to this problem uses a simple Markov chain, which we call the switch chain, to perform the…
We prove an invariance principle for non-stationary random processes and establish a rate of convergence under a new type of mixing condition. The dependence is exponentially decaying in the gap between the past and the future and is…
We study inhomogeneous continuous-time weakly ergodic Markov chains with a finite state space. We introduce the notion of a Markov chain with the regular structure of an infinitesimal matrix and study the sharp upper bounds on the rate of…
We consider the problem of estimating from sample paths the absolute spectral gap $\gamma_*$ of a reversible, irreducible and aperiodic Markov chain $(X_t)_{t \in \mathbb{N}}$ over a finite state space $\Omega$. We propose the ${\tt UCPI}$…
We give computable bounds on the rate of convergence of the transition probabilities to the stationary distribution for a certain class of geometrically ergodic Markov chains. Our results are different from earlier estimates of Meyn and…
We investigate the mixing properties of a model of reversible Markov chains in random environment, which notably contains the simple random walk on the superposition of a deterministic graph and a second graph whose vertex set has been…
The switch chain is a well-known Markov chain for sampling directed graphs with a given degree sequence. While not ergodic in general, we show that it is ergodic for regular degree sequences. We then prove that the switch chain is rapidly…
We investigate the sharpness of the spectral profile bound presented by Goel et al. and Chen et al. on the $L^{2}$ mixing time of Markov chains on continuous state spaces. We show that the bound provided by Chen et al. is sharp up to a…
We present a closed-form, computable expression for the expected number of times any transition event occurs during the transient phase of a reducible Markov chain. Examples of events include time to absorption, number of visits to a state,…
A new approach is developed for evaluating the convergence rate for nonlinear Markov chains (MC) based on the recently developed spectral radius technique of markovian coupling for linear MC and the idea of small nonlinear perturbations of…
It is well known that for a strictly stationary, reversible, Harris recurrent Markov chain, the $\rho$-mixing condition is equivalent to geometric ergodicity and to a "spectral gap" condition. In this note, it will be shown with an example…
We consider periodic Markov chains with absorption. Applying to iterates of this periodic Markov chain criteria for the exponential convergence of conditional distributions of aperiodic absorbed Markov chains, we obtain exponential…
In this paper, the question of expected time to convergence is addressed for unbiased quantized consensus on undirected connected graphs, and some strong results are obtained. The paper first provides a tight expression for the expected…
For a reversible and ergodic Markov chain $\{X_n,n\geq0\}$ with invariant distribution $\pi$, we show that a valid confidence interval for $\pi(h)$ can be constructed whenever the asymptotic variance $\sigma^2_P(h)$ is finite and positive.…
Let $(X_t)$ be a discrete time Markov chain on a general state space. It is well-known that if $(X_t)$ is aperiodic and satisfies a drift and minorization condition, then it converges to its stationary distribution $\pi$ at an exponential…