We provide a condition for f-ergodicity of strong Markov processes at a subgeometric rate. This condition is couched in terms of a supermartingale property for a functional of the Markov process. Equivalent formulations in terms of a drift inequality on the extended generator and on the resolvent kernel are given. Results related to (f,r)-regularity and to moderate deviation principle for integral (bounded) functional are also derived. Applications to specific processes are considered, including elliptic stochastic differential equation, Langevin diffusions, hypoelliptic stochastic damping Hamiltonian system and storage models.
The goal of this paper is to give a short and self contained proof of general bounds for subgeometric rates of convergence, under practical conditions. The main result whose proof, based on coupling, provides an intuitive understanding of the results of Nummelin and Tuominen (1983) and Tuominen and Tweedie (1994). To obtain practical rates, a very general drift condition, recently introduced in Douc et al (2004) is used.
It is well known that stationary geometrically ergodic Markov chains are $β$-mixing (absolutely regular) with geometrically decaying mixing coefficients. Furthermore, for initial distributions other than the stationary one, geometric ergodicity implies $β$-mixing under suitable moment assumptions. In this note we show that similar results hold also for subgeometrically ergodic Markov chains. In particular, for both stationary and other initial distributions, subgeometric ergodicity implies $β$-mixing with subgeometrically decaying mixing coefficients. Although this result is simple it should p…
In this paper we discuss how the notion of subgeometric ergodicity in Markov chain theory can be exploited to study stationarity and ergodicity of nonlinear time series models. Subgeometric ergodicity means that the transition probability measures converge to the stationary measure at a rate slower than geometric. Specifically, we consider suitably defined higher-order nonlinear autoregressions that behave similarly to a unit root process for large values of the observed series but we place almost no restrictions on their dynamics for moderate values of the observed series. Results on the subg…
In this paper, we provide sufficient conditions for the existence of the invariant distribution and for subgeometric rates of convergence in Wasserstein distance for general state-space Markov chains which are (possibly) not irreducible. Compared to previous work, our approach is based on a purely probabilistic coupling construction which allows to retrieve rates of convergence matching those previously reported for convergence in total variation. Our results are applied to establish the subgeometric ergodicity in Wasserstein distance of non-linear autoregressive models and of the pre-conditio…
We prove a central limit theorem for a general class of adaptive Markov Chain Monte Carlo algorithms driven by sub-geometrically ergodic Markov kernels. We discuss in detail the special case of stochastic approximation. We use the result to analyze the asymptotic behavior of an adaptive version of the Metropolis Adjusted Langevin algorithm with a heavy tailed target density.
A Markov chain is geometrically ergodic if it converges to its in- variant distribution at a geometric rate in total variation norm. We study geo- metric ergodicity of deterministic and random scan versions of the two-variable Gibbs sampler. We give a sufficient condition which simultaneously guarantees both versions are geometrically ergodic. We also develop a method for simul- taneously establishing that both versions are subgeometrically ergodic. These general results allow us to characterize the convergence rate of two-variable Gibbs samplers in a particular family of discrete bivariate di…
The aim of this note is to investigate the concentration properties of unbounded functions of geometrically ergodic Markov chains. We derive concentration properties of centered functions with respect to the square of the Lyapunov's function in the drift condition satisfied by the Markov chain. We apply the new exponential inequalities to derive confidence intervals for MCMC algorithms. Quantitative error bounds are providing for the regenerative Metropolis algorithm of [5].
We consider a stationary regularly varying time series which can be expressedas a function of a geometrically ergodic Markov chain. We obtain practical conditionsfor the weak convergence of the tail array sums and feasible estimators ofcluster statistics. These conditions include the so-called geometric drift or Foster-Lyapunovcondition and can be easily checked for most usual time series models witha Markovian structure. We illustrate these conditions on several models and statisticalapplications. A counterexample is given to show a different limiting behaviorwhen the geometric drift conditio…
The Horseshoe is a widely used and popular continuous shrinkage prior for high-dimensional Bayesian linear regression. Recently, regularized versions of the Horseshoe prior have also been introduced in the literature. Various Gibbs sampling Markov chains have been developed in the literature to generate approximate samples from the corresponding intractable posterior densities. Establishing geometric ergodicity of these Markov chains provides crucial technical justification for the accuracy of asymptotic standard errors for Markov chain based estimates of posterior quantities. In this paper, w…
In this paper we find nonasymptotic exponential upper bounds for the deviation in the ergodic theorem for families of homogeneous Markov processes. We find some sufficient conditions for geometric ergodicity uniformly over a parametric family. We apply this property to the nonasymptotic nonparametric estimation problem for ergodic diffusion processes.
Let $\{X_n\}_{n\ge0}$ be a $V$-geometrically ergodic Markov chain. Given some real-valued functional $F$, define $M_n(α):=n^{-1}\sum_{k=1}^nF(α,X_{k-1},X_k)$, $α\in\mathcal{A}\subset \mathbb {R}$. Consider an $M$ estimator $\hatα_n$, that is, a measurable function of the observations satisfying $M_n(\hatα_n)\leq \min_{α\in\mathcal{A}}M_n(α)+c_n$ with $\{c_n\}_{n\geq1}$ some sequence of real numbers going to zero. Under some standard regularity and moment assumptions, close to those of the i.i.d. case, the estimator $\hatα_n$ satisfies a Berry--Esseen theorem uniformly with respect to the under…
We describe ergodic properties of some Metropolis-Hastings (MH) algorithms for heavy-tailed target distributions. The analysis usually falls into sub-geometric ergodicity framework but we prove that the mixed preconditioned Crank-Nicolson (MpCN) algorithm has geometric ergodicity even for heavy-tailed target distributions. This useful property comes from the fact that the MpCN algorithm becomes a random-walk Metropolis algorithm under suitable transformation.
We state and prove a quantitative version of the bounded difference inequality for geometrically ergodic Markov chains. Our proof uses the same martingale decomposition as \cite{MR3407208} but, compared to this paper, the exact coupling argument is modified to fill a gap between the strongly aperiodic case and the general aperiodic case.
We study ergodic properties of some Markov chains models in random environments when the random Markov kernels that define the dynamic satisfy some usual drift and small set conditions but with random coefficients. In particular, we adapt a standard coupling scheme used for getting geometric ergodic properties for homogeneous Markov chains to the random environment case and we prove the existence of a process of randomly invariant probability measures for such chains, in the spirit of the approach of Kifer for chains satisfying some Doeblin type conditions. We then deduce ergodic properties of…
Bifurcating Markov chains (BMC) are Markov chains indexed by a full binary tree representing the evolution of a trait along a population where each individual has two children. We provide a central limit theorem for general additive functionals of BMC, and prove the existence of three regimes. This corresponds to a competition between the reproducing rate (each individual has two children) and the ergodicity rate for the evolution of the trait. This is in contrast with the work of Guyon (2007), where the considered additive functionals are sums of martingale increments, and only one regime app…
Strong invariance principles describe the error term of a Brownian approximation of the partial sums of a stochastic process. While these strong approximation results have many applications, the results for continuous-time settings have been limited. In this paper, we obtain strong invariance principles for a broad class of ergodic Markov processes. Strong invariance principles provide a unified framework for analysing commonly used estimators of the asymptotic variance in settings with a dependence structure. We demonstrate how this can be used to analyse the batch means method for simulation…
We prove a new concentration inequality for U-statistics of order two for uniformly ergodic Markov chains. Working with bounded and $π$-canonical kernels, we show that we can recover the convergence rate of Arcones and Gin{é} who proved a concentration result for U-statistics of independent random variables and canonical kernels. Our result allows for a dependence of the kernels $h_{i,j}$ with the indexes in the sums, which prevents the use of standard blocking tools. Our proof relies on an inductive analysis where we use martingale techniques, uniform ergodicity, Nummelin splitting and Bernst…
post.bgu.ac.il（BGU（可能是Ben-Gurion University of the Negev，具体需结合上下文））
We exhibit an efficient procedure for testing, based on a single long state sequence, whether an unknown Markov chain is identical to or $\varepsilon$-far from a given reference chain. We obtain nearly matching (up to logarithmic factors) upper and lower sample complexity bounds for our notion of distance, which is based on total variation. Perhaps surprisingly, we discover that the sample complexity depends solely on the properties of the known reference chain and does not involve the unknown chain at all, which is not even assumed to be ergodic.
Markov Chain Monte Carlo is repeatedly used to analyze the properties of intractable distributions in a convenient way. In this paper we derive conditions for geometric ergodicity of a general class of nonparametric stochastic volatility models with skewness driven by hidden Markov Chain with switching.
We investigate the statistical complexity of estimating the parameters of a discrete-state Markov chain kernel from a single long sequence of state observations. In the finite case, we characterize (modulo logarithmic factors) the minimax sample complexity of estimation with respect to the operator infinity norm, while in the countably infinite case, we analyze the problem with respect to a natural entry-wise norm derived from total variation. We show that in both cases, the sample complexity is governed by the mixing properties of the unknown chain, for which, in the finite-state case, there …
