"Let be a discrete or continuous-time Markov process with state space where is an arbitrary measurable set. Its transition semigroup is assumed to be additive with respect to the second component, i.e. is assumed to be a Markov additive process. In particular, this implies that the first component is also a Markov process. Markov random walks or additive functionals of a Markov process are special instances of Markov additive processes. In this paper, the process is shown to satisfy the following classical limit theorems:

(a) the central limit theorem,

(b) the local limit theorem,

(c) the one-dimensional Berry–Esseen theorem,

(d) the one-dimensional first-order Edgeworth expansion,

provided that we have with the expected order α with respect to the independent case (up to some ε > 0 for (c) and (d)). For the statements (b) and (d), a Markov nonlattice condition is also assumed as in the independent case. All the results are derived under the assumption that the Markov process has an invariant probability distribution π, is stationary and has the -spectral gap property (that is, (Xt)t∈ℕ is ρ-mixing in the discrete-time case). The case where is non-stationary is briefly discussed. As an application, we derive a Berry–Esseen bound for the M-estimators associated with ρ-mixing Markov chains."

"The multiple change-point problem is considered in the most general setting, where the only assumption made on the time-series distributions generating the data is that they are stationary ergodic. No modeling, independence or parametric assumptions are made. While the need for such a general setting is dictated by real applications, the problem of change-point estimation becomes a difficult unsupervised learning problem. In this work a novel algorithm for solving this problem is proposed, and it is shown to be asymptotically consistent under the general assumptions considered."

"The existence of invariant Borel probability measures for random dynamical systems on complete metric spaces is proved under assumptions that the systems have countably many maps and admit finite Markov partitions such that the resulting Markov systems are uniformly continuous and contractive, and satisfy some integrability condition in the infinite case. A one-to-one map between these measures and equilibrium states associated with such systems is established. Some properties of the map and the measures are given."

"In this note, we prove a conditionally centered version of the quenched weak invariance principle under the Hannan condition, for stationary processes. In the course, we obtain a (new) construction of the fact that any stationary process may be seen as a functional of a Markov chain."

"It is demonstrated how to represent asymptotically mean stationary (AMS) random sources with values in standard spaces as mixtures of ergodic AMS sources. This an extension of the well known decomposition of stationary sources which has facilitated the generalization of prominent source coding theorems to arbitrary, not necessarily ergodic, stationary sources. Asymptotic mean stationarity generalizes the definition of stationarity and covers a much larger variety of real-world examples of random sources of practical interest. It is sketched how to obtain source coding and related theorems for arbitrary, not necessarily ergodic, AMS sources, based on the presented ergodic decomposition."

"Two series of binary observations $x_1,x_1,...$ and $y_1,y_2,...$ are presented: at each time $ninN$ we are given $x_n$ and $y_n$. It is assumed that the sequences are generated independently of each other by two B-processes. We are interested in the question of whether the sequences represent a typical realization of two different processes or of the same one. We demonstrate that this is impossible to decide, in the sense that every discrimination procedure is bound to err with non-negligible frequency when presented with sequences from some B-processes. This contrasts earlier positive results on B-processes, in particular those showing that there are consistent $bar d$-distance estimates for this class of processes."

"In this paper we tackle the problem of fast rates in time series forecasting from a statistical learning perspective. In a serie of papers (e.g. Meir 2000, Modha and Masry 1998, Alquier and Wintenberger 2012) it is shown that the main tools used in learning theory with iid observations can be extended to the prediction of time series. The main message of these papers is that, given a family of predictors, we are able to build a new predictor that predicts the series as well as the best predictor in the family, up to a remainder of order $1/sqrt{n}$. It is known that this rate cannot be improved in general. In this paper, we show that in the particular case of the least square loss, and under a strong assumption on the time series (phi-mixing) the remainder is actually of order $1/n$. Thus, the optimal rate for iid variables, see e.g. Tsybakov 2003, and individual sequences, see cite{lugosi} is, for the first time, achieved for uniformly mixing processes. We also show that our method is optimal for aggregating sparse linear combinations of predictors."

--- Assumes observations are in the interval [-B,B] and gets a bound which is O(B^3), and so useless for our purposes.

"Computing smoothing distributions, the distributions of one or more states conditional on past, present, and future observations is a recurring problem when operating on general hidden Markov models. The aim of this paper is to provide a foundation of particle-based approximation of such distributions and to analyze, in a common unifying framework, different schemes producing such approximations. In this setting, general convergence results, including exponential deviation inequalities and central limit theorems, are established. In particular, time uniform bounds on the marginal smoothing error are obtained under appropriate mixing conditions on the transition kernel of the latent chain. In addition, we propose an algorithm approximating the joint smoothing distribution at a cost that grows only linearly with the number of particles."

"Static forecasting of stationary and ergodic time series is considered, i.e., inference of the conditional expectation of the response variable at time zero given the infinite past. It is shown that the mean squared error of a combination of suitably defined localized least squares estimates converges to zero for all distributions where the response variable is square integrable."

"Let $(X_t, Y_t)_{tin T}$ be a discrete or continuous-time Markov process with state space $X times R^d$ where $X$ is an arbitrary measurable set. Its transition semigroup is assumed to be additive with respect to the second component, i.e. $(X_t, Y_t)_{tin T}$ is assumed to be a Markov additive process. In particular, this implies that the first component $(X_t)_{tin T}$ is also a Markov process. Markov random walks or additive functionals of a Markov process are special instances of Markov additive processes. In this paper, the process $(Y_t)_{tin T}$ is shown to satisfy the following classical limit theorems: (a) the central limit theorem, (b) the local limit theorem, (c) the one-dimensional Berry-Esseen theorem, (d) the one-dimensional first-order Edgeworth expansion, provided that we have sup{tin(0,1]cap T : E{pi,0}[|Y_t| ^{alpha}] < 1 with the expected order with respect to the independent case (up to some $varepsilon > 0$ for (c) and (d)). For the statements (b) and (d), a Markov nonlattice condition is also assumed as in the independent case. All the results are derived under the assumption that the Markov process $(X_t)_{tin T}$ has an invariant probability distribution $pi$, is stationary and has the $L^2(pi)$-spectral gap property (that is, $(X_t)tin N}$ is $rho$-mixing in the discrete-time case). The case where $(X_t)_{tin T}$ is non-stationary is briefly discussed. As an application, we derive a Berry-Esseen bound for the M-estimators associated with $rho$-mixing Markov chains."

"The large-deviation method allows to characterize an ergodic counting process in terms of a thermodynamic frame where a free energy function determines the asymptotic non-stationary statistical properties of its fluctuations. Here, we study this formalism through a statistical mechanics approach, i.e., with an auxiliary counting process that maximizes an entropy function associated to the thermodynamic potential. We show that the realizations of this auxiliary process can be obtained after applying a conditional measurement scheme to the original ones, providing is this way an alternative measurement interpretation of the thermodynamic approach. General results are obtained for renewal counting processes, i.e., those where the time intervals between consecutive events are independent and defined by a unique waiting time distribution. The underlying statistical mechanics is controlled by the same waiting time distribution, rescaled by an exponential decay measured by the free energy function. A scale invariance, shift closure, and intermittence phenomena are obtained and interpreted in this context. Similar conclusions apply for non-renewal processes when the memory between successive events is induced by a stochastic waiting time distribution."

"Recent advances in single-molecule experiments show that various complex systems display nonergodic behavior. In this paper, we show how to test ergodicity and ergodicity breaking in experimental data. Exploiting the so-called dynamical functional, we introduce a simple test which allows us to verify ergodic properties of a real-life process. The test can be applied to a large family of stationary infinitely divisible processes. We check the performance of the test for various simulated processes and apply it to experimental data describing the motion of mRNA molecules inside live Escherichia coli cells. We show that the data satisfy necessary conditions for mixing and ergodicity. The detailed analysis is presented in the supplementary material."

"This article reviews some major episodes in the history of the spatial isomorphism problem of dynamical systems theory (ergodic theory). In particular, by analysing, both systematically and in historical context, a hitherto unpublished letter written in 1941 by John von Neumann to Stanislaw Ulam, this article clarifies von Neumann’s contribution to discovering the relationship between spatial isomorphism and spectral isomorphism. The main message of the article is that von Neumann’s argument described in his letter to Ulam is the very first proof that spatial isomorphism and spectral isomorphism are not equivalent because spectral isomorphism is weaker than spatial isomorphism: von Neumann shows that spectrally isomorphic ergodic dynamical systems with mixed spectra need not be spatially isomorphic."

"In this paper we present a Bernstein-type tail inequality for the maximum of partial sums of a weakly dependent sequence of random variables that is not necessarily bounded. The class considered includes geometrically and subgeometrically strongly mixing sequences. The result is then used to derive asymptotic moderate deviation results. Applications are given for classes of Markov chains, iterated Lipschitz models and functions of linear processes with absolutely regular innovations." Also: http://arxiv.org/abs/0902.0582

"We study the problem of parameter estimation for a univariate discretely observed ergodic diffusion process given as a solution to a stochastic differential equation. The estimation procedure we propose consists of two steps. In the first step, which is referred to as a smoothing step, we smooth the data and construct a nonparametric estimator of the invariant density of the process. In the second step, which is referred to as a matching step, we exploit a characterisation of the invariant density as a solution of a certain ordinary differential equation, replace the invariant density in this equation by its nonparametric estimator from the smoothing step in order to arrive at an intuitively appealing criterion function, and next define our estimator of the parameter of interest as a minimiser of this criterion function. In many interesting examples such an estimator will be computationally less intense than the more conventional estimators obtained through approximation of the likelihood function associated with the observations. Our main result shows that our estimator is $sqrt{n}$-consistent under suitable conditions. We also discuss a way of improving its asymptotic performance through a one-step Newton-Raphson type procedure."

Published version: J. Stat. Phys., 118 (2005): 555--588. Changes seem minor.

"In this paper we study ergodic properties of hidden Markov models with a generalized observation structure. In particular sufficient conditions for the existence of a unique invariant measure for the pair filter-observation are given. Furthermore, necessary and sufficient conditions for the existence of a unique invariant measure of the triple state-observation-filter are provided in terms of asymptotic stability in probability of incorrectly initialized filters. We also study the asymptotic properties of the filter and of the state estimator based on the observations as well as on the knowledge of the initial state. Their connection with minimal and maximal invariant measures is also studied."

"We describe conditions on non-gradient drift diffusion Fokker-Planck equations for its solutions to converge to equilibrium with a uniform exponential rate in Wasserstein distance. This asymptotic behaviour is related to a functional inequality, which links the distance with its dissipation and ensures a spectral gap in Wasserstein distance. We give practical criteria for this inequality and compare it to classical ones. The key point is to quantify the contribution of the diffusion term to the rate of convergence, which to our knowledge is a novelty."

"We study the generalization performance of arbitrary online learning algorithms trained on samples coming from a dependent source of data. We show that the generalization error of any stable online algorithm concentrates around its regret--an easily computable statistic of the online performance of the algorithm--when the underlying ergodic process is $beta$- or $phi$-mixing. We show high probability error bounds assuming the loss function is convex, and we also establish sharp convergence rates and deviation bounds for strongly convex losses and several linear prediction problems such as linear and logistic regression, least-squares SVM, and boosting on dependent data. In addition, our results have straightforward applications to stochastic optimization with dependent data, and our analysis requires only martingale convergence arguments; we need not rely on more powerful statistical tools such as empirical process theory."

"A new negative result for nonparametric distribution estimation of binary ergodic processes is shown. The problem of estimation of distribution with any degree of accuracy is studied. Then it is shown that for any countable class of estimators there is a zero-entropy binary ergodic process that is inconsistent with the class of estimators. Our result is different from other negative results for universal forecasting scheme of ergodic processes."

"Let H0 denote the class of all real valued i.i.d. processes and H1 all other ergodic real valued stationary processes. In spite of the fact that these classes are not countably tight we give a strongly consistent sequential test for distinguishing between them."

No, no, no, and again, no. Mainstream economics labors under many sins, but this harping on ergodicity is just confused. See http://bactra.org/weblog/679.html.

"We study the problem of estimating an unknown function from ergodic samples corrupted by additive noise. It is shown that one can consistently recover an unknown measurable function in this setting, if the one-dimensional (1-D) distribution of the samples is comparable to a known reference distribution, and the noise is independent of the samples and has known mixing rates. The estimates are applied to deterministic sampling schemes, in which successive samples are obtained by repeatedly applying a fixed map to a given initial vector, and it is then shown how the estimates can be used to reconstruct an ergodic transformation from one of its trajectories"

"A theorem of Ku\v{c}era states that given a Martin-L\"of random infinite binary sequence {\omega} and an effectively open set A of measure less than 1, some tail of {\omega} is not in A. We first prove several results in the same spirit and generalize them via an effective version of a weak form of Birkhoff's ergodic theorem. We then use this result to get a stronger form of it, namely a very general effective version of Birkhoff's ergodic theorem, which improves all the results previously obtained in this direction, in particular those of V'Yugin, Nandakumar and Hoyrup, Rojas."

"Dependence coefficients have been widely studied for Markov processes defined by a set of transition probabilities and an initial distribution. This work clarifies some aspects of the theory of dependence structure of Markov chains generated by copulas... relationship between the notions of geometric ergodicity and geometric {rho}-mixing ... for a large number of well known copulas, such as Clayton, Gumbel or Student, these notions are equivalent. Some of the results published in the last years appear to be redundant if one takes into account this fact. We apply this equivalence to show that any mixture of Clayton, Gumbel or Student copulas generate both geometrically ergodic and geometric {rho}-mixing stationary Markov chains, answering in this way an open question in the literature. We shall also point out that a sufficient condition for {rho}-mixing, used in the literature, actually implies Doeblin recurrence."

"The forward estimation problem for stationary and ergodic time series $\{X_n\}_{n=0}^{\infty}$ taking values from a finite alphabet ${\cal X}$ is to estimate the probability that $X_{n+1}=x$ based on the observations $X_i$, $0\le i\le n$ without prior knowledge of the distribution of the process $\{X_n\}$. We present a simple procedure $g_n$ which is evaluated on the data segment $(X_0,...,X_n)$ and for which, ${\rm error}(n) = |g_{n}(x)-P(X_{n+1}=x |X_0,...,X_n)|\to 0$ almost surely for a subclass of all stationary and ergodic time series, while for the full class the Cesaro average of the error tends to zero almost surely and moreover, the error tends to zero in probability."

"In this paper we derive the moderate deviation principle for stationary sequences of bounded random variables under martingale-type conditions. Applications to functions of $phi$-mixing sequences, contracting Markov chains, expanding maps of the interval, and symmetric random walks on the circle are given."

"We survey an area of recent development, relating dynamics to theoretical computer science. We discuss the theoretical limits of simulation and computation of interesting quantities in dynamical systems. We will focus on central objects of the theory of dynamics, as invariant measures and invariant sets, showing that even if they can be computed with arbitrary precision in many interesting cases, there exists some cases in which they can not. We also explain how it is possible to compute the speed of convergence of ergodic averages (when the system is known exactly) and how this entails the computation of arbitrarily good approximations of points of the space having typical statistical behaviour (a sort of constructive version of the pointwise ergodic theorem)."

"A concentration property of the functional $-\log f(X)$ is demonstrated, when a random vector $X$ has a log-concave density $f$ on $\R^n$. This concentration property implies in particular an extension of the Shannon-McMillan-Breiman strong ergodic theorem to the class of discrete-time stochastic processes with log-concave marginals."

"This paper presents a self-contained account for coupling arguments and applications in the context of Markov processes. We first use coupling to describe the transport problem, which leads to the concepts of optimal coupling and probability distance (or transportation-cost), then introduce applications of coupling to the study of ergodicity, Liouville theorem, convergence rate, gradient estimate, and Harnack inequality for Markov processes."

"A stationary random graph is a random rooted graph whose distribution is invariant under re-rooting along the simple random walk. We adapt the entropy technique developed for Cayley graphs and show in particular that stationary random graphs of subexponential growth are almost surely Liouville, that is, admit no non constant bounded harmonic function. Applications include the uniform infinite planar quadrangulation and long-range percolation clusters."

"Norbert Wiener really was that smart" dept.: long-term weather forecasting on the basis of deterministic dynamical models impossible because of limited precision observations and self-amplifying processes; but ergodic theory to the rescue for statistical forecasting; reconstruction of dynamical systems from sufficiently long trajectories (up to the ergodic component); linearization of nonlinear problems by projection into a higher-dimensional space; probably more, I'm not done reading it yet.

"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 to two examples: a two-component Gibbs sampler for the Normal distribution and a random logistic dynamical system."

Growing-order Markov approximations to the non-Markov ergodic process.  Assumes all predictive probabilities are > 0, and a sort of continuity property for the predictive distribution.  This rules out many interesting processes (e.g., even process) --- could proofs be modified?

"Various methods of summation for divergent series of real numbers have been generalized to analogous results for sums of i.i.d. random variables. The natural extension of results corresponding to Cesàro summation amounts to proving almost sure convergence of the Cesàro means. In the present paper we extend such results as well as weak laws and results on complete convergence to random fields, more specifically to random variables indexed by ℤ+2, the positive two-dimensional integer lattice points."

"We show that the sets in a family with finite VC dimension can be uniformly approximated within a given error by a finite partition. Immediate corollaries include the fact that VC classes have finite bracketing numbers, satisfy uniform laws of averages under strong dependence, and exhibit uniform mixing. Our results are based on recent work concerning uniform laws of averages for VC classes under ergodic sampling." I.e., they repeat their previous trick: if you don't have uniform approximation, they can construct arbitrarily large shatterable sets.

Sequel to their recent Annals of Probability paper, where they use the same trick to get convergence for function classes in terms of the gap (a.k.a. "fat shattering") dimension.

Oooh: "We show that if X is a complete separable metric space and C is a countable family of Borel subsets of with finite VC dimension, then, for every stationary ergodic process with values in X, the relative frequencies of sets c \in C converge uniformly to their limiting probabilities. Beyond ergodicity, no assumptions are imposed on the sampling process, and no regularity conditions are imposed on the elements of C. The result extends existing work of Vapnik and Chervonenkis, among others, who have studied uniform convergence for i.i.d. and strongly mixing processes. Our method of proof is new and direct: it does not rely on symmetrization techniques, probability inequalities or mixing conditions. The uniform convergence of relative frequencies for VC-major and VC-graph classes of functions under ergodic sampling is established as a corollary of the basic result for sets."  No rates, but very nice.

I really don't see how this abstract is compatible with Ryabko's own work!

"This informal introduction provides a fresh perspective on isomorphism theory, which is the branch of ergodic theory that explores the conditions under which two measure preserving systems are essentially equivalent. It contains a primer in basic measure theory, proofs of fundamental ergodic theorems, and material on entropy, martingales, Bernoulli processes, and various varieties of mixing. Original proofs of classic theorems - including the Shannon–McMillan–Breiman theorem, the Krieger finite generator theorem, and the Ornstein isomorphism theorem - are presented by degrees, together with helpful hints that encourage the reader to develop the proofs on their own. Hundreds of exercises and open problems are also included"

Very clever, though they do not derive explicit rates. --- I wonder if this couldn't be shown as a consequence of general properties of very-weak-Bernoulli processes? The vwB property would seem to imply perfect simulation, but I am not sure about the converse.

library has this but in off-site storage; request.
"There are many ways of introducing the concept of probability in classical, deterministic physics. This volume is concerned with one approach, known as 'the method of arbitrary functions', which was first considered by Poincare. ... proceeds by associating some uncertainty to our knowledge of both the initial conditions and the values of the physical constants that characterize the evolution of a physical system. By modeling this uncertainty by a probability density distribution ... analyze how the state of the system evolves through time. ... examples as diverse as bouncing balls, simple and coupled harmonic oscillators, integrable systems (such as spinning tops), planetary motion, and billiards. ... study the speed of convergence for solutions in order to determine the practical relevance of the method of arbitrary functions for specific examples. ... new results on convergence, and tractable upper bounds are derived"

"Recently some results have been presented which show that certain kinds of deterministic descriptions and indeterministic descriptions are observationally equivalent (Werndl 2009a, 2010). ... discuss the philosophical comments made by mathematicians about observational equivalence, in particular Ornstein and Weiss (1991). Their comments are vague, and I will argue that, according to a reasonable interpretation, they are misguided. Second, the results on observational equivalence raise the question of whether the deterministic or indeterministic description is preferable relative to all evidence [or it's underdetermined].... criticize Winnie's (1998) argument that, by appealing to different observations, one finds that the deterministic description is preferable. ... if the concern is a strong kind of underdetermination, the argument delivers the desired conclusion but this conclusion is trivial; and for other kinds of underdetermination ... the argument fails."

"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."

"A new negative result for nonparametric estimation of ergodic processes is shown. In this paper we restrict the class of estimators to computable functions and study estimation of distribution of ergodic processes with any given accuracy. Then we show a single ergodic process that is inconsistent with all computable estimators." --- But how is say the Ornstein-Weiss procedure incomputable? Must read carefully.

"We collect several observations that concern variable-length coding of two-sided infinite sequences in a probabilistic setting. Attention is paid to images and preimages of asymptotically mean stationary measures defined on subsets of these sequences. We point out sufficient conditions under which the variable-length coding and its inverse preserve asymptotic mean stationarity. Moreover, conditions for preservation of shift-invariant sigma-fields and the finite-energy property are discussed, and the block entropies for stationary means of coded processes are related in some cases. Subsequently, we apply certain of these results to construct a stationary nonergodic process with a desired linguistic interpretation."

"probabilities as deriving from ranges in suitably structured initial state spaces. Roughly, the probability of an event is the proportion of initial states that lead to this event in the space of all possible initial states, provided that this proportion is approximately the same in any not too small interval of the initial state space. This idea can also be expressed by saying that in the types of situations that give rise to probabilistic phenomena we may expect to find an initial state space such that any "reasonable" density function over this space leads to the same probabilities for the possible outcomes"

Yay, me! (6.5 years from project inception to publication. This is faster than self-organization stuff, which took 8 years.)

"We show that the conditional central limit theorem can take place for a stationary process defined on a nonergodic dynamical system while this last does not satisfy the central limit theorem for any ergodic component. There exists an ergodic Markov chain such that the conditional central limit theorem is satisfied for an invariant measure but fails to hold for almost all starting points."

"In this paper, a Hoeffding-type inequality is presented for a class of ergodic time series. The inequality is then used to construct uniformly exponentially consistent tests, which are useful tools for studying Bayesian consistency." Preprint at http://www4.stat.ncsu.edu/~sghosal/papers/Tang.pdf, haven't compared.

"We consider a partially hyperbolic set $K$ on a Riemannian manifold $M$ whose tangent space splits as $T_K M=E^{cu}\oplus E^{s}$, for which the centre-unstable direction $E^{cu}$ expands non-uniformly on some local unstable disk. We show that under these assumptions $f$ induces a Gibbs-Markov structure. Moreover, the decay of the return time function can be controlled in terms of the time typical points need to achieve some uniform expanding behavior in the centre-unstable direction. As an application of the main result we obtain certain rates for decay of correlations, large deviations, an almost sure invariance principle and the validity of the Central Limit Theorem."

"time-average variance constants" = sum of covariances = 1/correlation time (roughly).

"Dependence with complete connections is a more general type of stochastic process than the well-known Markovian dependence, accounting for a complete history of a stochastic evolution. This book is an authoritative survey of knowledge of the subject, dealing with the basic theoretical understanding and also with applications. These arise in a variety of situations as diverse as stochastic models of learning, branching processes in random environments, continued fractions and dynamical systems. Thus the book will appeal to mathematicians working in probability theory, ergodic theory and number theory, as well as applied mathematicians, engineers, biologists and social scientists interested in applications of stochastic methods." The "epsilon-machine" models I work with are an example of chains with complete connections.

"This paper concerns the statistical properties of hyperbolic diffeomorphisms. We obtain a large deviation result with respect to slowly shrinking intervals for a large class of Hölder continuous functions. In case of time reversal symmetry, we obtain a corresponding version of the Fluctuation Theorem."

I really wish I had taken this class when I had the chance.

Looks cool.

"The asymptotic normality of U-statistics has so far been proved for iid data and under various mixing conditions such as absolute regularity, but not for strong mixing. We use a coupling technique introduced in 1983 by Bradley to prove a new generalized covariance inequality similar to Yoshihara's. It follows from the Hoeffding-decomposition and this inequality that U-statistics of strongly mixing observations converge to a normal limit if the kernel of the U-statistic fulfills some moment and continuity conditions.
The validity of the bootstrap for U-statistics has until now only been established in the case of iid data (see Bickel and Freedman). For mixing data, Politis and Romano proposed the circular block bootstrap, which leads to a consistent estimation of the sample mean's distribution. We extend these results to U-statistics of weakly dependent data and prove a CLT for the circular block bootstrap version of U-statistics under absolute regularity and strong mixing. We also calculate a rate of convergence for the bootstrap variance estimator of a U-statistic and give some simulation results."

"We claim that looking at probability distributions of emph{finite time} largest Lyapunov exponents, and more precisely studying their large deviation properties, yields an extremely powerful technique to get quantitative estimates of polynomial decay rates of time correlations and Poincar'e recurrences in the -quite delicate- case of dynamical systems with weak chaotic properties."

"Given a discrete-valued sample X_1... X_n we wish to test whether it was generated by a process belonging to a family H_0, or it was generated by a process outside H_0. All process distributions are assumed stationary ergodic, and no further probabilistic or parametric assumptions are made. We require the Type I error of the test to be uniformly bounded, while the probability of Type II error has to tend to zero as the sample size increases. For this notion of consistency we provide necessary and sufficient conditions on the family H_0 for the existence of a consistent test. "

Determining the bias of a binary HMM: hard!

"A word-valued source $\mathbf{Y} = Y_1,Y_2,...$ is discrete random process that is formed by sequentially encoding the symbols of a random process $\mathbf{X} = X_1,X_2,...$ with codewords from a codebook $\mathscr{C}$. ... we prove the following: (1) if $\mathbf{X}$ is asymptotically mean stationary, then $\mathbf{Y}$ will satisfy a pointwise ergodic theorem and possess an AEP; and, (2) if the codebook $\mathscr{C}$ is prefix-free, then the entropy rate of $\mathbf{Y}$ is equal to the entropy rate of $\mathbf{X}$ normalized by the average codeword length." --- It's wonderful when things make sense like this.