Leonid Galtchouk, Serguei Pergamenchtchikov
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
Least squares estimation of the transition density in bifurcating Markov models
本文提出一种估计分叉马尔可夫模型转移密度的最小二乘方法，避免了核估计的商误差，并发展指数不等式证明其收敛速度依赖于遍历率，最后以数值研究验证理论结果。
Explicit Constraints on the Geometric Rate of Convergence of Random Walk Metropolis-Hastings
该研究细化拓展随机游走Metropolis-Hastings的几何遍历性充分条件，通过显式漂移与小集条件及谱理论得到收敛速度显式上下界，并应用于贝叶斯回归相关模型。
This short note reviews the basic theory for quantifying both the asymptotic and preasymptotic convergence of Markov chain Monte Carlo estimators.
We develop an effective strategy for proving strong ergodicity of (nonsymmetric) Markov semigroups associated to Hörmander type generators when the underlying configuration space is infinite dimensional.
We present Lyapunov-type conditions for non-strong ergodicity of Markov processes. Some concrete models are discussed including diffusion processes on Riemannian manifolds and Ornstein-Uhlenbeck processes driven by symmetric $α$-stable processes. For SDE driven by $α$-stable process ($α\in (0,2]$) with polynomial drift, the strong ergodicity or not is independent on $α$.
The Laplace transform of partial sums of the square of a non-centered Gauss-Markov process, conditioning on its starting point, is explicitly computed. The parameters of multiplicative ergodicity are deduced.
Ergodicity is a fundamental issue for a stochastic process. In this paper, we refine results on ergodicity for a general type of Markov chain to a specific type or the $GI/G/1$-type Markov chain, which has many interesting and important applications in various areas. It is of interest to obtain conditions in terms of system parameters or the given information about the process, under which the chain has various ergodic properties. Specifically, we provide necessary and sufficient conditions for geometric, strong and polynomial ergodicity, respectively.
In the present paper we investigate the $L_1$-weak ergodicity of nonhomogeneous discrete Markov processes with general state spaces. Note that the $L_1$-weak ergodicity is weaker than well-known weak ergodicity. We provide a necessary and sufficient condition for such processes to satisfy the $L_1$-weak ergodicity. Moreover, we apply the obtained results to establish $L_1$-weak ergodicity of discrete time quadratic stochastic processes. As an application of the main result, certain concrete examples are also provided.
Non-(strong, geometrically) ergodicity criteria for discrete time Markov chains on general state
本文提出了一般状态上离散时间马尔可夫链非强几何ergodic性的判据，通过最小非负解理论推导出充分条件，并应用于特定单出生死亡过程的非ergodic性分析。
Inverse Problems for Ergodicity of Markov Chains
本文针对连续与离散时间马尔可夫链，通过建立基于 $Q$-矩阵或转移矩阵不等式的可解性判据，解决了普通、代数、指数和强遍历性的反问题，并展示了其在单生过程及多维模型中的通用应用。
We consider a time inhomogeneous strong Markov process $(ξ_t)_{t\ge 0}$ taking values in a Polish state space whose semigroup has a $T$-periodic structure. We give simple conditions which imply ergodicity of the grid chain $(ξ_{kT})_{k\in \mathbb{N}_0}$ In case of $d$-dimensional possibly degenerate diffusions, the conditions are stated in terms of drift and diffusion coefficient of the process.
We prove the convergence at an exponential rate towards the invariant probability measure for a class of solutions of stochastic differential equations with finite delay. This is done, in this non-Markovian setting, using the cluster expansion method, inspired from [4] or [14]. As a consequence, the results hold for small perturbations of ergodic diffusions.
In this paper the stability and the perturbation bounds of Markov operators acting on abstract state spaces are investigated. Here, an abstract state space is an ordered Banach space where the norm has an additivity property on the cone of positive elements. We basically study uniform ergodic properties of Markov operators by means of so-called a generalized Dobrushin's ergodicity coefficient. This allows us to get several convergence results with rates. Some results on quasi-compactness of Markov operators are proved in terms of the ergodicity coefficient. Furthermore, a characterization of u…
We study properties of the Laplace transforms of non-negative additive functionals of Markov chains. We are namely interested in a multiplicative ergodicity property used in [18] to study bifurcating processes with ancestral dependence. We develop a general approach based on the use of the operator perturbation method. We apply our general results to two examples of Markov chains, including a linear autoregressive model. In these two examples the operator-type assumptions reduce to some expected finite moment conditions on the functional (no exponential moment conditions are assumed in this wo…
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…
We derive sufficient conditions for subgeometric f-ergodicity of strongly Markovian processes. We first propose a criterion based on modulated moment of some delayed return-time to a petite set. We then formulate a criterion for polynomial f-ergodicity in terms of a drift condition on the generator. Applications to specific processes are considered, including Langevin tempered diffusions on R^n and storage models.
We study perturbation theory and uniform ergodicity for discrete-time Markov chains on general state spaces in terms of the uniform moments of the first hitting times on some set. The methods we adopt are different from previous ones. For reversible and non-negative definite Markov chains, we first investigate the geometrically ergodic convergence rates. Based on the estimates, together with a first passage formula, we then get the convergence rates in uniform ergodicity. If the transition kernel $P$ is only reversible, we transfer to study the two-skeleton chain with the transition kernel $P^…
Generalization of the Lorden's inequality is an excellent tool for obtaining strong upper bounds for the convergence rate for various complicated stochastic models. This paper demonstrates a method for obtaining such bounds for some generalization of the Markov modulated Poisson process (MMPP). The proposed method can be applied to reliability and queuing theory.
We study coercive inequalities in Orlicz spaces associated to the probability measures on finite and infinite dimensional spaces which tails decay slower than the Gaussian ones. We provide necessary and sufficient criteria for such inequalities to hold and discuss relations between various classes of inequalities.
We are interested in quasi-stationarity and quasi-ergodicity when the absorbing boundary is moving. First we show that, in the moving boundary case, the quasi-stationary distribution and the quasi-limiting distribution are not well-defined when the boundary is oscillating periodically. Then we show the existence of a quasi-ergodic distribution for any discrete-time irreducible Markov chain defined on a finite space state in the fixed boundary case. Finally we use this last result to show the quasi-ergodicity in the moving boundary case.
In our earlier paper, a generalized Dobrushin ergodicity coefficient of Markov operators (acting on abstract state spaces) with respect to a projection $P$, has been introduced and studied. It turned out that the introduced coefficient was more effective than the usual ergodicity coefficient. In the present work, by means of a left consistent Markov projections and the generalized Dobrushin's ergodicity coefficient, we investigate uniform and weak $P$-ergodicities of non-homogeneous discrete Markov chains (NDMC) on abstract state spaces. It is easy to show that uniform $P$-ergodicity implies a…
The particle Gibbs (PG) sampler is a systematic way of using a particle filter within Markov chain Monte Carlo (MCMC). This results in an off-the-shelf Markov kernel on the space of state trajectories, which can be used to simulate from the full joint smoothing distribution for a state space model in an MCMC scheme. We show that the PG Markov kernel is uniformly ergodic under rather general assumptions, that we will carefully review and discuss. In particular, we provide an explicit rate of convergence which reveals that: (i) for fixed number of data points, the convergence rate can be made ar…
A Route to Harris ergodicity for Non-Feller Markov Kernels
针对满足拟Feller分解的非Feller马尔可夫核，在ψ不可约性下，通过紧-小集准则得到小紧集，结合强制Lyapunov函数和Hairer-Mattingly压缩定理，在几何漂移条件下证明加权全变差范数下的几何遍历性。
In this paper we prove a sharp quantitative version of the Kendall's Theorem. The Kendal Theorem states that under some mild conditions imposed on a probability distribution on positive integers (i.e. probabilistic sequence) one can prove convergence of its renewal sequence. Due to the well-known property - the first entrance last exit decomposition - such results are of interest in the stability theory of time homogeneous Markov chains. In particular the approach may be used to measure rates of convergence of geometrically ergodic Markov chains and consequently implies estimates on convergenc…
This note extends the work of Foss and Tweedie (1997), who showed that availability of the classic Coupling from The Past algorithm of Propp and Wilson (1996) is essentially equivalent to uniform ergodicity for a Markov chain (see also HobertRobert, 2004). In this note we show that all geometrically ergodic chains possess dominated Coupling from The Past algorithms (not necessarily practical!) which are rather closely connected to Foster-Lyapunov criteria.
We apply Doeblin's ergodicity coefficient as a computational tool to approximate the occupancy distribution of a set of states in a homogeneous but possibly non-stationary finite Markov chain. Our approximation is based on new properties satisfied by this coefficient, which allow us to approximate a chain of duration n by independent and short-lived realizations of an auxiliary homogeneous Markov chain of duration of order ln(n). Our approximation may be particularly useful when exact calculations via first-step methods or transfer matrices are impractical, and asymptotic approximations may no…
We consider general Markov chains with discrete time in an arbitrary measurable (phase) space and homogeneous in time. Markov chains are defined by the classical transition function which within the framework of the operator treatment generates a conjugate pair of linear Markov operators in the Banach space of measurable bounded functions and in the Banach space of bounded finite additive measures. It is proved that the well-known Doeblin condition $ (D) $ of ergodicity (quasi\-compactness) of the Markov chain is equivalent to the condition $ (*) $: all finitely additive invariant measures of …
Max-stable processes are central models for spatial extremes. In this paper, we focus on some space-time max-stable models introduced in Embrechts et al. (2016). The processes considered induce discrete-time Markov chains taking values in the space of continuous functions from the unit sphere of $\mathbb{R}^3$ to $(0, \infty)$. We show that these Markov chains are geometrically ergodic. An interesting feature lies in the fact that the state space is not locally compact, making the classical methodology inapplicable. Instead, we use the fact that the state space is Polish and apply results pres…
We are interested in the dynamic of a structured branching population where the trait of each individual moves according to a Markov process. The rate of division of each individual is a function of its trait and when a branching event occurs, the trait of a descendant at birth depends on the trait of the mother. We prove a law of large numbers for the empirical distribution of ancestral trajectories. It ensures that the empirical measure converges to the mean value of the spine which is a time-inhomogeneous Markov process describing the trait of a typical individual along its ancestral lineag…
We study the exponential growth of bifurcating processes with ancestral dependence. We suppose here that the lifetimes of the cells are dependent random variables, that the numbers of new cells are random and dependent. Lifetimes and new cells's numbers are also assumed to be dependent. We illustrate our results by examples, including some Markov models. Our approach is related to the behaviour of the Laplace transform of nonnegative additive functional of Markov chains and require weak moment assumption (no exponential moment is needed).
Ergodicity of inhomogeneous Markov processes under general criteria
本文针对非齐次马尔可夫过程，通过推广Krylov-Bogolyubov方法或采用Lyapunov准则建立不变测度族的存在性，并在压缩假设下证明唯一性与指数遍历性，以三个实例验证方法的实用性。
In this work, we consider an inhomogeneous (discrete time) Markov chain and are interested in its long time behavior. We provide sufficient conditions to ensure that some of its asymptotic properties can be related to the ones of a homogeneous (continuous time) Markov process. Renowned examples such as a bandit algorithms, weighted random walks or decreasing step Euler schemes are included in our framework. Our results are related to functional limit theorems, but the approach differs from the standard "Tightness/Identification" argument; our method is unified and based on the notion of pseudo…
Equivalences of Geometric Ergodicity of Markov Chains
本文系统整理了时间齐次马尔可夫链几何遍历性的34个等价条件，涵盖收敛界、漂移条件和谱性质等，并给出了条件间联系的证明。
Motivated by a model presented by S. Gudder, we study a quantum generalization of Markov chains and discuss the relation between these maps and open quantum random walks, a class of quantum channels described by S. Attal et al. We consider processes which are nonhomogeneous in time, i.e., at each time step, a possibly distinct evolution kernel. Inspired by a spectral technique described by L. Saloff-Coste and J. Zúñiga, we define a notion of ergodicity for nonhomogeneous quantum Markov chains and describe a criterion for ergodicity of such objects in terms of singular values. As a consequence …
Ergodicity for eventually continuous Markov--Feller semigroups on Polish spaces
本文研究波兰空间上马尔可夫--费勒半群在极弱正则性条件下的遍历性，证明了Cesàro平均的弱收敛性，刻画了Cesàro最终连续性、e-性质与弱-*平均遍历性的关系，并给出了不变测度存在唯一性准则及遍历分解性质，最后应用于多个非平凡系统。
Asymptotic behavior of the three-dimensional stochastic Navier-Stokes equations with Markov switching in additive noises is studied for incompressible fluid flow in a bounded domain in the three-dimensional space. To study such a system, we introduce a family of regularized equations and investigate the asymptotic behavior of the regularized equations first. The existence an ergodic measure for the regularized system is established via the Krylov-Bogolyubov method. Then the existence of an stationary measure to the original system is obtained by extracting a limit from the ergodic measures of …
Imprecise Markov Semigroups and their Ergodicity
本文提出不精确马尔可夫半群的概念，通过拓扑、几何和概率方法分析模型不确定性下的平稳极限，研究了长期界限的统一性和渐进行为中不精确性的消失。
Let $P$ be a Markov kernel on a measurable space $\X$ and let $V:\X\r[1,+\infty)$. This paper provides explicit connections between the $V$-geometric ergodicity of $P$ and that of finite-rank nonnegative sub-Markov kernels $\Pc_k$ approximating $P$. A special attention is paid to obtain an efficient way to specify the convergence rate for $P$ from that of $\Pc_k$ and conversely. Furthermore, explicit bounds are obtained for the total variation distance between the $P$-invariant probability measure and the $\Pc_k$-invariant positive measure. The proofs are based on the Keller-Liverani perturbat…
We establish quantitative bounds for rates of convergence and asymptotic variances for iterated conditional sequential Monte Carlo (i-cSMC) Markov chains and associated particle Gibbs samplers. Our main findings are that the essential boundedness of potential functions associated with the i-cSMC algorithm provide necessary and sufficient conditions for the uniform ergodicity of the i-cSMC Markov chain, as well as quantitative bounds on its (uniformly geometric) rate of convergence. Furthermore, we show that the i-cSMC Markov chain cannot even be geometrically ergodic if this essential boundedn…
This paper studies the f-ergodicity and its exponential convergence rate for continuous-time Markov chain. Assume f is square integrable, for reversible Markov chain, it is proved that the exponential convergence of f-ergodicity holds if and only if the spectral gap of the generator is positive. Moreover, the convergence rate is equal to the spectral gap. For irreversible case, the positivity of spectral gap remains a sufficient condition of f-ergodicity. The effectiveness of these results are illustrated by some typical examples.
A path method for non-exponential ergodicity of Markov chains and its application for chemical reaction systems
本文提出通过路径转移速率比验证马尔可夫链非指数遍历性的准则，应用于生化反应网络，发现环路反应主导可引发非指数遍历性，并构造了缓慢收敛的平衡生化系统。
We study ergodic properties of a class of Markov-modulated general birth-death processes under fast regime switching. The first set of results concerns the ergodic properties of the properly scaled joint Markov process with a parameter that is taken large. Under very weak hypotheses, we show that if the averaged process is exponentially ergodic for large values of the parameter, then the same applies to the original joint Markov process. The second set of results concerns steady-state diffusion approximations, under the assumption that the 'averaged' fluid limit exists. Here, we establish conv…
Let $P$ be a Markov kernel on a measurable space $\X$ and let $V:\X\r[1,+\infty)$. We provide various assumptions, based on drift conditions, under which $P$ is quasi-compact on the weighted-supremum Banach space $(\cB_V,\|\cdot\|_V)$ of all the measurable functions $f : \X\r\C$ such that $\|f\|_V := \sup_{x\in \X} |f(x)|/V(x) < \infty$. Furthermore we give bounds for the essential spectral radius of $P$. Under additional assumptions, these results allow us to derive the convergence rate of $P$ on $\cB_V$, that is the geometric rate of convergence of the iterates $P^n$ to the stationary distri…
The goal of this paper is to develop a general method to establish conditional ergodicity of infinite-dimensional Markov chains. Given a Markov chain in a product space, we aim to understand the ergodic properties of its conditional distributions given one of the components. Such questions play a fundamental role in the ergodic theory of nonlinear filters. In the setting of Harris chains, conditional ergodicity has been established under general nondegeneracy assumptions. Unfortunately, Markov chains in infinite-dimensional state spaces are rarely amenable to the classical theory of Harris cha…
Exponential ergodicity in the bounded-Lipschitz distance for a subclass of piecewise-deterministic Markov processes with random switching between flows
本文研究了一类在流之间随机切换的分段确定性马尔可夫过程，提出了仅依赖流性质和跳跃后位置转移律的可验证条件，以保证其在有界Lipschitz距离下的指数遍历性，并为适用于生物模型的特例建立了简单判据。
Exponential ergodicity of some Markov dynamical system with application to a Poisson driven stochastic differential equation
研究状态相关跳跃强度的分段确定性 Markov 过程的渐近性，证明其唯一不变分布在有界 Lipschitz 对偶距离中指数吸引，并推广了 Poisson 驱动随机微分方程跳跃过程的结果。
We formulate a criterion for the existence and uniqueness of an invariant measure for a Markov process taking values in a Polish phase space. In addition, weak-$^*$ ergodicity, that is, the weak convergence of the ergodic averages of the laws of the process starting from any initial distribution, is established. The principal assumptions are the existence of a lower bound for the ergodic averages of the transition probability function and its local uniform continuity. The latter is called the e-property. The general result is applied to solutions of some stochastic evolution equations in Hilbe…
It is known that the Dobrushin's ergodicity coefficient is one of the effective tools to study a behavior of non-homogeneous Markov chains. In the present paper, we define such an ergodicity coefficient of a positive mapping defined on ordered Banach space with a base (OBSB), and study its properties. In terms of this coefficient we prove the equivalence uniform and weak ergodicities of homogeneous Markov chains. This extends earlier results obtained in case of von Neumann algebras. Such a result allowed to establish a category theorem for uniformly ergodic Markov operators. We find necessary …
Exponential ergodicity and finite-dimensional approximation for Markovian lifts of stochastic Volterra equations
本文通过广义Harris定理建立了随机Volterra方程马尔可夫提升的指数ergodic性，并证明了其平稳解可由有限维SDEs弱近似。
The paper is dedicated to studying the problem of existence and uniqueness of solutions as well as existence of and exponential convergence to invariant measures for McKean-Vlasov stochastic differential equations with Markovian switching. Since the coefficients are only locally Lipschitz, we need to truncate them both in space and distribution variables simultaneously to get the global existence of solutions under the Lyapunov condition. Furthermore, if the Lyapunov condition is strengthened, we establish the exponential convergence of solutions' distributions to the unique invariant measure …
We study the ergodic behaviour of a discrete-time process $X$ which is a Markov chain in a stationary random environment. The laws of $X_t$ are shown to converge to a limiting law in (weighted) total variation distance as $t\to\infty$. Convergence speed is estimated and an ergodic theorem is established for functionals of $X$. Our hypotheses on $X$ combine the standard "small set" and "drift" conditions for geometrically ergodic Markov chains with conditions on the growth rate of a certain "maximal process" of the random environment. We are able to cover a wide range of models that have hereto…
Uniform ergodicity of geodesic slice sampling
本文证明了测地线切片采样在紧支撑且有界密度分布上的均匀遍历性，并显式给出了收敛常数及其对超参数、目标分布和定义域的依赖关系。
We consider a bivariate stationary Markov chain $(X_n,Y_n)_{n\ge0}$ in a Polish state space, where only the process $(Y_n)_{n\ge0}$ is presumed to be observable. The goal of this paper is to investigate the ergodic theory and stability properties of the measure-valued process $(Π_n)_{n\ge0}$, where $Π_n$ is the conditional distribution of $X_n$ given $Y_0,...,Y_n$. We show that the ergodic and stability properties of $(Π_n)_{n\ge0}$ are inherited from the ergodicity of the unobserved process $(X_n)_{n\ge0}$ provided that the Markov chain $(X_n,Y_n)_{n\ge0}$ is nondegenerate, that is, its trans…
We develop a general framework for studying ergodicity of order-preserving Markov semigroups. We establish natural and in a certain sense optimal conditions for existence and uniqueness of the invariant measure and exponential convergence of transition probabilities of an order-preserving Markov process. As an application, we show exponential ergodicity and exponentially fast synchronization-by-noise of the stochastic reaction-diffusion equation in the hypoelliptic setting. This refines and complements corresponding results of Hairer, Mattingly (2011).
We develop necessary and sufficient conditions for uniqueness of the invariant measure of the filtering process associated to an ergodic hidden Markov model in a finite or countable state space. These results provide a complete solution to a problem posed by Blackwell (1957), and subsume earlier partial results due to Kaijser, Kochman and Reeds. The proofs of our main results are based on the stability theory of nonlinear filters.
We study a Markov process with two components: the first component evolves according to one of finitely many underlying Markovian dynamics, with a choice of dynamics that changes at the jump times of the second component. The second component is discrete and its jump rates may depend on the position of the whole process. Under regularity assumptions on the jump rates and Wasserstein contraction conditions for the underlying dynamics, we provide a concrete criterion for the convergence to equilibrium in terms of Wasserstein distance. The proof is based on a coupling argument and a weak form of …
Generally the convergence rate in exponential ergodicity $λ$ is an upper bound for the convergence rate $κ$ in uniform ergodicity for a Markov process, that is $λ\geqslantκ$. In this paper, we prove that $κ\geqslant \inf \{lambda,1/M_H\}$, where $M_H$ is a uniform bound on the moment of the hitting time to a "compact" set $H$. In the case where $M_H$ can be made arbitrarily small for $H$ large enough, we obtain that $λ=κ$. The general results are applied to Markov chains, diffusion processes and solutions to SDEs driven by symmetric stable processes.
Foundations of locally-balanced Markov processes
本文引入并研究一般状态空间上的局部平衡马尔可夫跳跃过程（LBMJPs），解决离散参数及非光滑分布采样局限，确立其适定性、遍历性等性质，推导指数遍历条件与扩散极限，拓展蒙特卡洛采样应用。
We consider a Markovian single-server retrial queueing system with a constant retrial rate. Conditions of null ergodicity and exponential ergodicity for the correspondent process, as well as bounds on the rate of convergence are obtained.
We establish verifiable general sufficient conditions for exponential or subexponential ergodicity of Markov processes that may lack the strong Feller property. We apply the obtained results to show exponential ergodicity of a variety of nonlinear stochastic partial differential equations with additive forcing, including 2D stochastic Navier-Stokes equations. Our main tool is a new version of the generalized coupling method.
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。