Limit Theorem For Markov Chains Research Articles

A quantum-mechanical derivation of the multivariate central limit theorem for Markov chains

Abstract The central limit theorem for Markov chains is widely used, in particular in its pristine univariate form. As far as the multivariate case is concerned, a few proofs exist, which depend on different assumptions and require advanced mathematical and statistical tools. Here a novel proof is presented that, starting from the standard condition of regularity only, relies on time-independent quantum-mechanical perturbation theory. The result, which is obtained by using techniques that are typical of physics, is expected to enhance the usability of this cornerstone theorem, especially in nonlinear dynamics and physics of complex systems.

Ergodicity and central limit theorem for random interval homeomorphisms

The central limit theorem for Markov chains generated by iterated function systems consisting of orientation-preserving homeomorphisms of the interval is proved. We study also ergodicity of such systems.

General Bernstein-Like Inequality for Additive Functionals of Markov Chains

Using the renewal approach, we prove Bernstein-like inequalities for additive functionals of geometrically ergodic Markov chains, thus obtaining counterparts of inequalities for sums of independent random variables. The coefficient in the sub-Gaussian part of our estimate is the asymptotic variance of the additive functional, i.e., the variance of the limiting Gaussian variable in the central limit theorem for Markov chains. This refines earlier results by Adamczak and Bednorz, obtained under the additional assumption of strong aperiodicity of the chain.

Some generalized strong limit theorems for Markov chains in bi-infinite random environments

In this paper, we study a strong limit theorem of delayed sums for Markov chains in bi-infinite environments by constructing a sequence of random variables with parameters. As a corollary, we obtain some strong limit properties including the generalized conditional relative entropy for Markov chains in bi-infinite random environment. The results which we obtained generalize the results of Liu et al. (2015).

Local stationarity and time-inhomogeneous Markov chains

A primary motivation of this contribution is to define new locally stationary Markov models for categorical or integer-valued data. For this initial purpose, we propose a new general approach for dealing with time-inhomogeneity that extends the local stationarity notion developed in the time series literature. We also introduce a probabilistic framework which is very flexible and allows us to consider a much larger class of Markov chain models on arbitrary state spaces, including most of the locally stationary autoregressive processes studied in the literature. We consider triangular arrays of time-inhomogeneous Markov chains, defined by some families of contracting and slowly-varying Markov kernels. The finite-dimensional distribution of such Markov chains can be approximated locally with the distribution of ergodic Markov chains and some mixing properties are also available for these triangular arrays. As a consequence of our results, some classical geometrically ergodic homogeneous Markov chain models have a locally stationary version, which lays the theoretical foundations for new statistical modeling. Statistical inference of finite-state Markov chains can be based on kernel smoothing and we provide a complete and fast implementation of such models, directly usable by the practitioners. We also illustrate the theory on a real data set. A central limit theorem for Markov chains on more general state spaces is also provided and illustrated with the statistical inference in INAR models, Poisson ARCH models and binary time series models. Additional examples such as locally stationary regime-switching or SETAR models are also discussed.

A functional limit theorem for coin tossing Markov chains

Nous prouvons un theoreme limite fonctionnelle pour les chaines de Markov qui, a chaque etape, montent ou descendent avec probabilite 1/2 d’une constante dependante de l’etat. Le theoreme implique que la loi de chaque processus de Markov uni-dimensionel, fort, continu, regulier et a l’echelle naturelle peut etre approximee par de telles chaines de Markov avec precision quelconque. Le theoreme limite fonctionnelle s’applique en particulier aux processus de Markov qui ne peuvent pas etre caracterises comme solutions d’une equation differentielle stochastique. Notamment nos resultats permettent d’approximer de tels processus avec un comportement irregulier; nous illustrons cela avec des processus de Markov «collants», par exemple, le mouvement brownien «collant» et un mouvement brownien ralenti sur l’ensemble de Cantor.

The Central Limit Theorem for Markov Chains with General State Space

We consider a Markov chain with general state space and an embedded Markov chain sampled at the times of successive returns to a subsetA0 of the state space.We assume that the latter chain is uniformly ergodic but the originalMarkov chain need not possess this property.We develop amodification of the spectralmethod and utilize it in proving the central limit theorem for theMarkov chain under consideration.

Persistence of sums of correlated increments and clustering in cellular automata

Let T be the first return time to (−∞,0] of sums of increments given by a functional of a stationary Markov chain. We determine the asymptotic behavior of the survival probability, P(T≥t)∼Ct−1∕2 for an explicit constant C. Our analysis is based on a connection between the survival probability and the running maximum of the time-reversed process, and relies on a functional central limit theorem for Markov chains. As applications, we recover known clustering results for the 3-color cyclic cellular automaton and the Greenberg–Hastings model, and we prove a new clustering result for the 3-color firefly cellular automaton.

Markov Chain Monte Carlo confidence intervals

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. We do not impose any additional condition on the convergence rate of the Markov chain. The confidence interval is derived using the so-called fixed-b lag-window estimator of $\sigma_P^2(h)$. We also derive a result that suggests that the proposed confidence interval procedure converges faster than classical confidence interval procedures based on the Gaussian distribution and standard central limit theorems for Markov chains.

Bootstrap uniform central limit theorems for Harris recurrent Markov chains

The main objective of this paper is to establish bootstrap uniform functional central limit theorem for Harris recurrent Markov chains over uniformly bounded classes of functions. We show that the result can be generalized also to the unbounded case. To avoid some complicated mixing conditions, we make use of the well-known regeneration properties of Markov chains. We show that in the atomic case the proof of the bootstrap uniform central limit theorem for Markov chains for functions dominated by a function in $L^{2}$ space proposed by Radulović (2004) can be significantly simplified. Finally, we prove bootstrap uniform central limit theorems for Fréchet differentiable functionals in a Markovian setting.

Limit theorems for Markov chains by the symmetrization method

Let P be a Markov operator with invariant probability m, ergodic on L2(S,m), and let (Wn)n≥0 be the Markov chain with state space S and transition probability P on the space of trajectories (Ω,Pm), with initial distribution m. Following Wu and Olla we define the symmetrized operator Ps=(P+P⁎)/2, and analyze the linear manifold H−1:=I−PsL2(S,m). We obtain for real f∈H−1 an explicit forward–backward martingale decomposition with a coboundary remainder. For such f we also obtain some maximal inequalities for Sn(f):=∑k=0nf(Wk), related to the law of iterated logarithm. We prove an almost sure central limit theorem for f∈H−1 when P is normal in L2(S,m), or when P satisfies the sector condition. We characterize the sector condition by the numerical range of P on the complex L2(S,m) being in a sector with vertex at 1. We then show that if P has a real normal dilation which satisfies the sector condition, then H−1=I−PL2(S,m). We use our approach to prove that P is L2-uniformly ergodic if and only if it satisfies (the discrete) Poincaré's inequality.

A strong limit theorem for the average of ternary functions of Markov chains in bi-infinite random environments

We consider strong limit theorems for Markov chains in bi-infinite random environments. We first give a new proof of a strong limit theorem of Liu and Yang (1995) on the average of functions of non-homogeneous Markov chains by constructing a nonnegative martingale. As corollaries, for a Markov chain in a bi-infinite random environment, we obtain strong limit theorems for the conditional relative entropy and for the number of times that the Markov chain and related processes reach a given point.

An almost sure local limit theorem for Markov chains

An almost sure local limit theorem for Markov chains is investigated.

Variance estimation in the central limit theorem for Markov chains

This article concerns the variance estimation in the central limit theorem for finite recurrent Markov chains. The associated variance is calculated in terms of the transition matrix of the Markov chain. We prove the equivalence of different matrix forms representing this variance. The maximum likelihood estimator for this variance is constructed and it is proved that it is strongly consistent and asymptotically normal. The main part of our analysis consists in presenting closed matrix forms for this new variance. Additionally, we prove the asymptotic equivalence between the empirical and the maximum likelihood estimation (MLE) for the stationary distribution.

Bounds on regeneration times and limit theorems for subgeometric Markov chains

Nous établissons dans ce papier des théorèmes limites pour des chaînes de Markov à espace d’état général sous des conditions impliquant l’ergodicité sous géométrique. Sous des conditions de dérive et de minorisation plus faibles que celles de Foster–Lyapounov, nous obtenons un théorème de limite centrale et un principe de déviation modérée pour des fonctionnelles additives non nécessairement bornées de la chaîne de Markov. La preuve repose sur la méthode de régénération et un contrôle précis du moment modulé de temps d’atteinte d’ensembles petits.

On Variance Conditions for Markov Chain CLTs

Central limit theorems for Markov chains are considered, and in particular the relationships between various expressions for asymptotic variance known from the literature. These turn out to be equal under fairly general conditions, although not always. We also investigate the existence of CLTs, and pose some open problems.

A Note on the Martingale Method of Proving the Central Limit Theorem for Stationary Sequences

Under appropriate assumptions, the martingale approximation method allows us to reduce the study of the asymptotic behavior of sums of random variables that form a stationary random sequence to a similar problem for sums of stationary martingale differences. In an early paper on the martingale method, the author have proposed certain sufficient conditions for the central limit theorem to hold. It is shown in the present note that these conditions, at least in one particular case, can be essentially relaxed. In the context of the central limit theorem for Markov chains, a similar observation was done in a recent Holzmann and author's work. Bibliography: 12 titles.

Central limit theorems for contractive Markov chains

We prove limit theorems for Markov chains under (local) contraction conditions. As a corollary we obtain a central limit theorem for Markov chains associated with iterated function systems with contractive maps and place-dependent Dini-continuous probabilities.

The Markov approximation of the random fields on Cayley trees and a class of small deviation theorems

By introducing the sample relative entropy rate as a measure of the deviation between the arbitrary random fields and the Markov chain fields on Cayley trees, a class of small deviation theorems for the frequencies of state ordered couples is established. In the proof a new analytic technique in the study of the strong limit theorems for Markov chains is applied.

The central limit theorem for Markov chains started at a point

The aim of this paper is to prove a central limit theorem and an invariance principle for an additive functional of an ergodic Markov chain on a general state space, with respect to the law of the chain started at a point. No irreducibility assumption nor mixing conditions are imposed; the only assumption bears on the growth of the L 2 -norms of the ergodic sums for the function generating the additive functional, which must be with . The result holds almost surely with respect to the invariant probability of the chain.