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 Bernstein's type inequality. Assuming further that the Markov chain starts from its invariant distribution, we prove a Bernstein-type concentration inequality that provides sharper convergence rate for small variance terms.
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 give a bound on the mixing time of a uniformly ergodic, reversible Markov chain in terms of the spectral radius of the transition operator. This bound has been established previously in finite state spaces, and is widely believed to hold in general state spaces, but a proof has not been provided to our knowledge.
In this paper a concentration inequality is proved for the deviation in the ergodic theorem in the case of discrete time observations of diffusion processes. The proof is based on the geometric ergodicity property for diffusion processes. As an application we consider the nonparametric pointwise estimation problem for the drift coefficient under discrete time observations.
In this article, three models are considered, they are the infinitely-many-neutral-alleles model \cite{MR615945}, infinite dimensional diffusion associated with two-parameter Poisson-Dirichlet distribution \cite{MR2596654} and the infinitely-many-alleles model with symmetric dominance \cite{MR1626158}. The new representations of the transition transition densities are obtained for the first two models. Lastly, the ergodic inequalities of these three models are provided.
In this note, we present a version of Hoeffding's inequality in a continuous-time setting, where the data stream comes from a uniformly ergodic diffusion process. Similar to the well-studied case of Hoeffding's inequality for discrete-time uniformly ergodic Markov chain, the proof relies on techniques ranging from martingale theory to classical Hoeffding's lemma as well as the notion of deviation kernel of diffusion process. We present two examples to illustrate our results. In the first example we consider large deviation probability on the occupation time of the Jacobi diffusion, a popular p…
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.
We prove an isoperimetric inequality for the uniform measure on a uniformly convex body and for a class of uniformly log-concave measures (that we introduce). These inequalities imply (up to universal constants) the log-Sobolev inequalities proved by Bobkov--Ledoux as well as the isoperimetric inequalities due to Bakry-Ledoux and Bobkov--Zegarlinski. We also recover a concentration inequality for uniformly convex bodies, similar to that proved by Gromov--Milman.
Let $f$ be a martingale with values in a uniformly $p$-smooth Banach space and $w$ any positive weight. We show that $\mathbb{E} (f^* \cdot w) \lesssim \mathbb{E}(S_p f \cdot w^*)$, where $\cdot^*$ is the martingale maximal operator and $S_p$ is the $\ell^p$ sum of martingale increments.
In this paper, we prove an inequality, which we call "Devroye inequality", for a large class of non-uniformly hyperbolic dynamical systems (M,f). This class, introduced by L.-S. Young, includes families of piece-wise hyperbolic maps (Lozi-like maps), scattering billiards (e.g., planar Lorentz gas), unimodal and H{é}non-like maps. Devroye inequality provides an upper bound for the variance of observables of the form K(x,f(x),...,f^{n-1}(x)), where K is any separately Holder continuous function of n variables. In particular, we can deal with observables which are not Birkhoff averages. We will s…
We prove that an irreducible aperiodic Markov chain is geometrically ergodic if and only if any separately bounded functional of the stationary chain satisfies an appropriate subgaussian deviation inequality from its mean.
This paper surveys the main results obtained during the period 1992-1999 on three aspects mentioned at the title. The first result is a new and general variational formula for the lower bound of spectral gap (i.e., the first non-trivial eigenvalue) of elliptic operators in Euclidean space, Laplacian on Riemannian manifolds or Markov chains (§1). Here, a probabilistic method-coupling method is adopted. The new formula is a dual of the classical variational formula. The last formula is actually equivalent to Poincaré inequality. To which, there are closely related logarithmic Sobolev inequality,…
We study ergodic properties of stochastic dissipative systems with additive noise. We show that the system is uniformly exponentially ergodic provided the growth of nonlinearity at infinity is faster than linear. The abstract result is applied to the stochastic reaction diffusion equation in $\mathbb R^d$ with $d\le 3$.
We use here a particle system to prove a convergence result as well as a deviation inequality for solutions of granular media equation when the confinement potential and the interaction potential are no more uniformly convex. Proof is straightforward, simplifying deeply proofs of Carrillo-McCann-Villani \cite{CMV,CMV2} and completing results of Malrieu \cite{malrieu03} in the uniformly convex case. It relies on an uniform propagation of chaos property and a direct control in Wasserstein distance of solutions starting with different initial measures. The deviation inequality is obtained via a $…
A first-order Liouville theorem is obtained for random ensembles of uniformly parabolic systems under the mere qualitative assumptions of stationarity and ergodicity. Furthermore, the paper establishes, almost surely, an intrinsic large-scale $C^{1,α}$-regularity estimate for caloric functions.
In this paper, we apply Devroye inequality to study various statistical estimators and fluctuations of observables for processes. Most of these observables are suggested by dynamical systems. These applications concern the co-variance function, the integrated periodogram, the correlation dimension, the kernel density estimator, the speed of convergence of empirical measure, the shadowing property and the almost-sure central limit theorem. We proved in \cite{CCS} that Devroye inequality holds for a class of non-uniformly hyperbolic dynamical systems introduced in \cite{young}. In the second app…
We consider ergodic backward stochastic differential equations in a discrete time setting, where noise is generated by a finite state Markov chain. We show existence and uniqueness of solutions, along with a comparison theorem. To obtain this result, we use a Nummelin splitting argument to obtain ergodicity estimates for a discrete time Markov chain which hold uniformly under suitable perturbations of its transition matrix. We conclude with an application of this theory to a treatment of an ergodic control problem.
We investigate the properties of a discrete-time martingale $\{X_m\}_{m\in \mathbb Z_{\geq 0}}$, where all differences between adjacent random variables are limited to be not more than a constant as a promise. In this situation, it is known that the Azuma-Hoeffding inequality holds, which gives an upper bound of a probability for exceptional events. The inequality gives a simple form of the upper bound, and it has been utilized for many investigations. However, the inequality is not tight. We give an explicit expression of a tight upper bound, and we show that it and the bound obtained from th…
The first aim of the present note is to quantify the speed of convergence of a conditioned process toward its Q-process under suitable assumptions on the quasi-stationary distribution of the process. Conversely, we prove that, if a conditioned process converges uniformly to a conservative Markov process which is itself ergodic, then it admits a unique quasi-stationary distribution and converges toward it exponentially fast, uniformly in its initial distribution. As an application, we provide a conditional ergodic theorem.
We prove that ergodic measures on one-sided shift spaces are uniformly scaling in the sense of Gavish. That is, given a shift ergodic measure we prove that at almost every point the scenery distributions weakly converge to a common distribution on the space of measures. Moreover, we give an explicit description of the limiting distribution in terms of a `reverse Jacobian' function associated with the corresponding measure on the space of left infinite sequences.
We consider ergodic backward stochastic differential equations, in a setting where noise is generated by a countable state uniformly ergodic Markov chain. We show that for Lipschitz drivers such that a comparison theorem holds, these equations admit unique solutions. To obtain this result, we show by coupling and splitting techniques that uniform ergodicity estimates of Markov chains are robust to perturbations of the rate matrix, and that these perturbations correspond in a natural way to EBSDEs. We then consider applications of this theory to Markov decision problems with a risk-averse avera…
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.
In this article, we establish Hoeffding's inequality for bounded Lipschitz functions of a class of not necessarily irreducible Markov models. The result complements the existing literature on this topic where Hoeffding's inequality for bounded measurable functions of a class of irreducible Markov models has been considered. Our approach is based on the assumption of uniform ergodicity of the underlying Markov model in $L^1$-Wasserstein space.
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…
In this paper, a fourth moment bound for partial sums of functional of strongly ergodic Markov chain is established. This type of inequality plays an important role in the study of empirical process invariance principle. This one is specially adapted to the technique of Dehling, Durieu and Volný (2008). The same moment bound can be proved for dynamical system whose transfer operator has some spectral properties. Examples of applications are given.
We present a tail inequality for suprema of empirical processes generated by variables with finite $ψ_α$ norms and apply it to some geometrically ergodic Markov chains to derive similar estimates for empirical processes of such chains, generated by bounded functions. We also obtain a bounded difference inequality for symmetric statistics of such Markov chains.
