We give sufficient conditions for central limit theorems (CLTs) for additive functionals of Markov chains in terms of quantitative Wasserstein convergence rates, including suitable subgeometric rates. For a given metric $ψ$, we establish CLTs for $ψ$-Lipschitz functions under moment conditions by showing that suitable $1$-Wasserstein convergence rates imply either the Maxwell--Woodroofe projective criterion or convergence of the associated Poisson series. We then extend this framework beyond the $ψ$-Lipschitz setting in two directions. First, by reweighting $ψ$ with a non-negative function $V$, we construct a weighted path metric under which functions with $V$-controlled increments are Lipschitz. We derive convergence bounds in the Wasserstein distance induced by this new metric from corresponding bounds in the Wasserstein distance induced by $ψ$, thereby obtaining CLTs for this broad class of functions. Second, we consider functions admitting integrable increment envelopes and derive CLTs from quantitative $2$-Wasserstein convergence rates. On $\mathbb R^d$, pointwise Sobolev inequalities and polynomial bounds on the gradient provide concrete sufficient conditions for constructing such envelopes. We illustrate the results with nonlinear autoregressive processes and a random walk on the one-dimensional torus exhibiting subgeometric Wasserstein convergence.
The goal of this paper is to describe conditions which guarantee a central limit theorem for random variables, which distributions are controled by hidden Markov chains. We proved that when a Markov chain is ergodic and random variables fullfiled Lindeberg's condition then the Central Limit Theorem is true.
Our purpose is to prove central limit theorem for countable nonhomogeneous Markov chain under the condition of uniform convergence of transition probability matrices for countable nonhomogeneous Markov chain in Cesàro sense. Furthermore, we obtain a corresponding moderate deviation theorem for countable nonhomogeneous Markov chain by Gärtner-Ellis theorem and exponential equivalent method.
We present a framework for obtaining explicit bounds on the rate of convergence to equilibrium of a Markov chain on a general state space, with respect to both total variation and Wasserstein distances. For Wasserstein bounds, our main tool is Steinsaltz's convergence theorem for locally contractive random dynamical systems. We describe practical methods for finding Steinsaltz's "drift functions" that prove local contractivity. We then use the idea of "one-shot coupling" to derive criteria that give bounds for total variation distances in terms of Wasserstein distances. Our methods are applied…
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…
The purpose of this paper is to extend the investigation of Poisson-type deviation inequalities started by Joulin (Bernoulli 13 (2007) 782--798) to the empirical mean of positively curved Markov jump processes. In particular, our main result generalizes the tail estimates given by Lezaud (Ann. Appl. Probab. 8 (1998) 849--867, ESAIM Probab. Statist. 5 (2001) 183--201). An application to birth--death processes completes this work.
We consider the problem of statistical inference in a parametric finite Markov chain model and develop a robust estimator of the parameters defining the transition probabilities via minimization of a suitable (empirical) version of the popular density power divergence. Based on a long sequence of observations from a first-order stationary Markov chain, we have defined the minimum density power divergence estimator (MDPDE) of the underlying parameter and rigorously derived its asymptotic and robustness properties under appropriate conditions. Performance of the MDPDEs is illustrated theoretical…
We consider a Markov chain on $\mathbb{R}^d$ with invariant measure $μ$. We are interested in the rate of convergence of the empirical measures towards the invariant measure with respect to various dual distances, including in particular the $1$-Wasserstein distance. The main result of this article is a new upper bound for the expected distance, which is proved by combining a Fourier expansion with a truncation argument. Our bound matches the known rates for i.i.d. random variables up to logarithmic factors. In addition, we show how concentration inequalities around the mean can be obtained.
Markov chain models are used in various fields, such behavioral sciences or econometrics. Although the goodness of fit of the model is usually assessed by large sample approximation, it is desirable to use conditional tests if the sample size is not large. We study Markov bases for performing conditional tests of the toric homogeneous Markov chain model, which is the envelope exponential family for the usual homogeneous Markov chain model. We give a complete description of a Markov basis for the following cases: i) two-state, arbitrary length, ii) arbitrary finite state space and length of thr…
We derive a Markov basis consisting of moves of degree at most three for two-state toric homogeneous Markov chain model of arbitrary length without parameters for initial states. Our basis consists of moves of degree three and degree one, which alter the initial frequencies, in addition to moves of degree two and degree one for toric homogeneous Markov chain model with parameters for initial states.
Let $μ_N$ be the empirical measure associated to a $N$-sample of a given probability distribution $μ$ on $\mathbb{R}^d$. We are interested in the rate of convergence of $μ_N$ to $μ$, when measured in the Wasserstein distance of order $p>0$. We provide some satisfying non-asymptotic $L^p$-bounds and concentration inequalities, for any values of $p>0$ and $d\geq 1$. We extend also the non asymptotic $L^p$-bounds to stationary $ρ$-mixing sequences, Markov chains, and to some interacting particle systems.
We study the problem of non-asymptotic deviations between a reference measure and its empirical version, in the 1-Wasserstein metric, under the standing assumption that the measure satisfies a transport-entropy inequality. We extend some results of F. Bolley, A. Guillin and C. Villani with simple proofs. Our methods are based on concentration inequalities and extend to the general setting of measures on a Polish space. Deviation bounds for the occupation measure of a Markov chain are also given, under the assumption that the chain is contractive on the space of Lipschitz functions. Throughout …
We establish a simple variance inequality for U-statistics whose underlying sequence of random variables is an ergodic Markov Chain. The constants in this inequality are explicit and depend on computable bounds on the mixing rate of the Markov Chain. We apply this result to derive the strong law of large number for U-statistics of a Markov Chain under conditions which are close from being optimal.
We consider dependence coefficients for stationary Markov chains. We emphasize on some equivalencies for reversible Markov chains. We improve some known results and provide a necessary condition for Markov chains based on Archimedean copulas to be exponential $ρ$-mixing. We analyze the example of the Mardia and Frechet copula families using small sets.
We prove that moderate deviations for empirical measures for countable nonhomogeneous Markov chains hold under the assumption of uniform convergence of transition probability matrices for countable nonhomogeneous Markov chains in Cesàro sense.
This paper brings some insights of $ψ'$-mixing, $ψ^*$-mixing and $ψ$-mixing for copula-based Markov chains and the perturbations of their copulas. We provide new tools to check Markov chains for $ψ$-mixing or $ψ'$-mixing, and also show that perturbations of $ψ'$-mixing copula-based Markov chains are $ψ'$-mixing while perturbations of $ψ$-mixing Markov chains are not necessarily $ψ$-mixing Markov chains, even when the perturbed copula is $ψ-mixing$. Some examples of copula families are considered. A statistical study is provided to emphasize the impact of perturbations on copula-based Markov ch…
The efficiency of a Markov sampler based on the underdamped Langevin diffusion is studied for high dimensional targets with convex and smooth potentials. We consider a classical second-order integrator which requires only one gradient computation per iteration. Contrary to previous works on similar samplers, a dimension-free contraction of Wasserstein distances and convergence rate for the total variance distance are proven for the discrete time chain itself. Non-asymptotic Wasserstein and total variation efficiency bounds and concentration inequalities are obtained for both the Metropolis adj…
For a Markov chain both the detailed balance condition and the cycle Kolmogorov condition are algebraic binomials. This remark suggests to study reversible Markov chains with the tool of Algebraic Statistics, such as toric statistical models. One of the results of this study in an algebraic parameterization of reversible Markov transitions and their invariant probability.
We prove that the three-state toric homogenous Markov chain model has Markov degree two. In algebraic terminology this means, that a certain class of toric ideals are generated by quadratic binomials. This was conjectured by Haws, Martin del Campo, Takemura and Yoshida, who proved that they are generated by binomials of degree six or less.
