In this paper we survey some recent results on the central limit theorem and its weak invariance principle for stationary sequences. We also describe several maximal inequalities that are the main tool for obtaining the invariance principles, and also they have interest in themselves. The classes of dependent random variables considered will be martingale-like sequences, mixing sequences, linear processes, additive functionals of ergodic Markov chains.
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.
The classical Donsker weak invariance principle is extended to a Besov spaces framework. Polygonal line processes build from partial sums of stationary martingale differences as well independent and identically distributed random variables are considered. The results obtained are shown to be optimal.
We provide some sufficient mixing conditions on a strictly stationary sequence in order to guarantee the weak invariance principle in Hölder spaces. Strong mixing and $ρ$-mixing conditions are investigated as well as $τ$-dependent sequences. The main tools are Fuk-Nagaev type inequalities for mixing sequences and a truncation argument.
In 1983, N. Herrndorf proved that for a $ϕ$-mixing sequence satisfying the central limit theorem and $\liminf_{n\to\infty}\frac{σ^2_n}n>0$, the weak invariance principle takes place. The question whether for strictly stationary sequences with finite second moments and a weaker type ($α$, $β$, $ρ$) of mixing the central limit theorem implies the weak invariance principle remained open. We construct a strictly stationary $β$-mixing sequence with finite moments of any order and linear variance for which the central limit theorem takes place but not the weak invariance principle.
We derive a new maximal inequality for stationary sequences under a martingale-type condition introduced by Maxwell and Woodroofe [Ann. Probab. 28 (2000) 713-724]. Then, we apply it to establish the Donsker invariance principle for this class of stationary sequences. A Markov chain example is given in order to show the optimality of the conditions imposed.
On the weak invariance principle for strictly stationary sequences in some Besov spaces
Université de Strasbourg（斯特拉斯堡大学）；CNRS（法国国家科学研究中心）；V. I. Romanovskiy Institute of Mathematics, Uzbekistan Academy of Sciences（乌兹别克斯坦科学院 V.I.罗曼诺夫斯基数学研究所）
We study a symmetric diffusion $X$ on $\mathbb{R}^d$ in divergence form in a stationary and ergodic environment, with measurable unbounded and degenerate coefficients $a^ω$. The diffusion is formally associated with $L^ωu = \nabla\cdot(a^ω\nabla u)$, and we make sense of it through Dirichlet forms theory. We prove for $X$ a quenched invariance principle, under some moment conditions on the environment; the key tool is the sublinearity of the corrector obtained by Moser's iteration scheme.
In this note we (in particular) prove an almost sure invariance principle (ASIP) for non-stationary and uniformly bounded sequences of random variables which are exponentially fast $ϕ$-mixing. The obtained rate is of order $o(V_n^{\frac14+\del})$ for an arbitrary $\del>0$, where $V_n$ is the variance of the underlying partial sums $S_n$. For certain classes of inhomogeneous Markov chains we also prove a vector-valued ASIP with similar rates.
General Central limit theorem deals with weak limits (in type) of sums of row-elements of array random variables. In some situations as in the invariance principle problem, the sums may include only parts of the row-elements. For strictly stationary arrays (stationary for each row), there is no change to the asymptotic results. But for non-stationary data, especially for dependent data, asymptotic laws of partial sums moving in rows may require extra-conditions to exist. This paper deals with central limit theorems with Gaussian limits for non-stationary data. Our main focus is on dependent da…
We obtain a necessary and sufficient condition for the orthomartingale-coboundary decomposition. We establish a sufficient condition for the approximation of the partial sums of a strictly stationary random fields by those of stationary orthomartingale differences. This condition can be checked under multidimensional analogues of the Hannan condition and the Maxwell-Woodroofe condition.
We establish a central limit theorem and an invariance principle for stationary random fields, with projective-type conditions. Our result is obtained via an m-dependent approximation method. As applications, we establish invariance principles for orthomartingales and functionals of linear random fields.
We investigate the invariance principle in H{ö}lder spaces for strictly stationary martingale difference sequences. In particular, we show that the sufficient condition on the tail in the i.i.d. case does not extend to stationary ergodic martingale differences. We provide a sufficient condition on the conditional variance which guarantee the invariance principle in H{ö}lder spaces. We then deduce a condition in the spirit of Hannan one.
We establish an invariance principle for a general class of stationary random fields indexed by $\mathbb Z^d$, under Hannan's condition generalized to $\mathbb Z^d$. To do so we first establish a uniform integrability result for stationary orthomartingales, and second we establish a coboundary decomposition for certain stationary random fields. At last, we obtain an invariance principle by developing an orthomartingale approximation. Our invariance principle improves known results in the literature, and particularly we require only finite second moment.
We establish a central limit theorem for partial sums of stationary linear random fields with dependent innovations, and an invariance principle for anisotropic fractional Brownian sheets. Our result is a generalization of the invariance principle for fractional Brownian motions by Dedecker et al. (2011) to high dimensions. A key ingredient of their argument, the martingale approximation, is replaced by an m-approximation argument. An important tool of our approach is a moment inequality for stationary random fields recently established by El Machkouri et al. (2011).
The simple Lévy Poisson process and scaled forms are explicitly constructed from partial sums of independent and identically distributed random variables and from sums of non-stationary independent random variables. For the latter, the weak limits are scaled Poisson processes. The method proposed here prepares generalizations to dependent data, to associated data in the first place.
We propose certain conditions which are sufficient for the functional law of the iterated logarithm (the Strassen invariance principle) for some general class of non-stationary Markov-Feller chains. This class may be briefly specified by the following two properties: firstly, the transition operator of the chain under consideration enjoys a non-linear Lyapunov-type condition, and secondly, there exists an appropriate Markovian coupling whose transition probability function can be decomposed into two parts, one of which is contractive and dominant in some sense. The construction of such a coupl…
In this paper we survey the almost sure central limit theorem and its functional form (quenched) for stationary and ergodic processes. For additive functionals of a stationary and ergodic Markov chain these theorems are known under the terminology of central limit theorem and its functional form, started at a point. All these results have in common that they are obtained via a martingale approximation in the almost sure sense. We point out several applications of these results to classes of mixing sequences, shift processes, reversible Markov chains, Metropolis Hastings algorithms.
We prove a strong invariance principle for the Kantorovich distance between the empiricaldistribution and the marginal distribution of stationary $α$-mixing sequences.
We investigate the invariance principle for set-indexed partial sums of a stationary field $(X\_{k})\_{k\in\mathbb{Z}^{d}}$ of martingale-difference or independent random variables under standard-normalization or self-normalization respectively.
In this paper we estimate the rest of the approximation of a stationary process by a martingale in terms of the projections of partial sums. Then, based on this estimate, we obtain almost sure approximation of partial sums by a martingale with stationary differences. The results are exploited to further investigate the central limit theorem and its invariance principle started at a point, as well as the law of the iterated logarithm via almost sure approximation with a Brownian motion, improving the results available in the literature. The conditions are well suited for a variety of examples; …
The aim of this article is to refine a weak invariance principle for stationary sequences given by Doukhan & Louhichi (1999). Since our conditions are not causal our assumptions need to be stronger than the mixing and causal $θ$-weak dependence assumptions used in Dedecker & Doukhan (2003). Here, if moments of order $>2$ exist, a weak invariance principle and convergence rates in the CLT are obtained; Doukhan & Louhichi (1999) assumed the existence of moments with order $>4$. Besides the previously used $η$- and $κ$-weak dependence conditions, we introduce a weaker one, $λ$, which fits the Ber…
We prove a sequence of limiting results about weakly dependent stationary and regularly varying stochastic processes in discrete time. After deducing the limiting distribution for individual clusters of extremes, we present a new type of point process convergence theorem. It is designed to preserve the entire information about the temporal ordering of observations which is typically lost in the limit after time scaling. By going beyond the existing asymptotic theory, we are able to prove a new functional limit theorem. Its assumptions are satisfied by a wide class of applied time series models…
The results of Komlós, Major and Tusnády give optimal Wiener approximation of partial sums of i.i.d. random variables and provide an extremely powerful tool in probability and statistical inference. Recently Wu [Ann. Probab. 35 (2007) 2294--2320] obtained Wiener approximation of a class of dependent stationary processes with finite $p$th moments, $2<p\le4$, with error term $o(n^{1/p}(\log n)^γ)$, $γ>0$, and Liu and Lin [Stochastic Process. Appl. 119 (2009) 249--280] removed the logarithmic factor, reaching the Komlós--Major--Tusnády bound $o(n^{1/p})$. No similar results exist for $p>4$, and i…
The investigation asymptotic limits on associated data mainly focused on limit theorems of summands of associated data and on the related invariance principles. In a series of papers, we are going to set the general frame of the theory by considering an arbitrary infinitely decomposable (divisible) limit law for summands and study the associated functional laws converging to Lévy processes. The asymptotic frame of Newman (1980) is still used as a main tool. Detailed results are given when $G$ is a Gaussian law (as confirmation of known results) and when $G$ is a Poisson law. In the later case,…
In this paper, we study the asymptotic behavior of the normalized cadlag functions generated by the discrete Fourier transforms of a stationary centered square-integrable process, started at a point. We prove that the quenched invariance principle holds for averaged frequencies under no assumption other than ergodicity, and that this result holds also for almost every fixed frequency under a certain generalization of the Hannan condition and a certain rotated form of the Maxwell and Woodroofe condition which, under a condition of weak dependence that we specify, is guaranteed for a.e. frequenc…
For a class of stationary regularly varying and weakly dependent time series, we prove the so-called complete convergence result for the corresponding space-time point processes. As an application of our main theorem, we give a simple proof of the invariance principle for the corresponding partial maximum process.
We consider empirical processes generated by strictly stationary sequences of associated random variables. S. Louhichi established an invariance principle for such processes, assuming that the covariance function decays rapidly enough. We show that under certain conditions imposed on the pairwise distributions of the random variables in question the restrictions on the rate of decay of the covariance function can be relaxed.
We find a sufficient condition under which a central limit theorem for a stationary linear process is quenched. We find a stationary linear process szatisfying the Maxwell-Woodroofe condition for which the variances of partial sums are o(n), there is a CLT with a convergence towards N(0,1) when dividing by standard deviation of the partial sums, and the CLT is not quenched. The weak invariance principle does not hold.
We establish an invariance principle for a one-dimensional random walk in a dynamical random environment given by a speed-change exclusion process. The jump probabilities of the walk depend on the configuration of the exclusion in a finite box around the walker. The environment starts from equilibrium. After a suitable space-time rescaling, the random walk converges to a sum of two independent processes, a Brownian motion and a Gaussian process with stationary increments.
We study a continuous time random walk X in an environment of dynamic random conductances. We assume that the conductances are stationary ergodic, uniformly bounded and bounded away from zero and polynomially mixing in space and time. We prove a quenched invariance principle for X, and obtain Green's functions bounds and a local limit theorem. We also discuss a connection to stochastic interface models.
We establish strong invariance principles for sums of stationary and ergodic processes with nearly optimal bounds. Applications to linear and some nonlinear processes are discussed. Strong laws of large numbers and laws of the iterated logarithm are also obtained under easily verifiable conditions.
We consider the process of partial sums of moving averages of finite order with a regular varying memory function, constructed from a stationary sequence, variance of the sum of which is a regularly varying function. We study the Gaussian approximation of this process of partial sums with the aid of a certain class of Gaussian processes, and obtain sufficient conditions for the $C$-convergence in the invariance principle in the Donsker form
Inspired by \citet{Berkes14} and \citet{Wu07}, we prove an almost sure invariance principle for stationary $β-$mixing stochastic processes defined on Hilbert space. Our result can be applied to Markov chain satisfying Meyn-Tweedie type Lyapunov condition and thus generalises the contraction condition in \citet[Example 2.2]{Berkes14}. We prove our main theorem by the big and small blocks technique and an embedding result in \citet{gotze2011estimates}. Our result is further applied to the ergodic Markov chain and functional autoregressive processes.
Let $(X_i)_{i\in\Z}$ be a regular stationary process for a given filtration. The weak invariance principle holds under the condition $\sum_{i\in\Z}\|P_0(X_i)\|_2<\infty$ (see Hannan (1979)}, Dedecker and Merlevède (2003), Deddecker, Merlevéde and Volný (2007)). In this paper, we show that this criterion is independent of other known criteria: the martingale-coboundary decomposition of Gordin (see Gordin (1969, 1973)), the criterion of Dedecker and Rio (see Dedecker and Rio (2000)) and the condition of Maxwell and Woodroofe (see Maxwell and Woodroofe (2000), Peligrade and Utev (2005), Volný (20…
We give rates of convergence in the strong invariance principle for stationary sequences satisfying some projective criteria. The conditions are expressed in terms of conditional expectations of partial sums of the initial sequence. Our results apply to a large variety of examples, including mixing processes of different kinds. We present some applications to symmetric random walks on the circle, to functions of dependent sequences, and to a reversible Markov chain.
We consider the simple random walk on random graphs generated by discrete point processes. This random graph has a random subset of a cubic lattice as the vertices and lines between any consecutive vertices on lines parallel to each coordinate axis as the edges. Under the assumption that discrete point processes are finitely dependent and stationary, we prove that the quenched invariance principle holds, that is, for almost every configuration of a point process, the path distribution of the walk converges weakly to that of a Brownian motion.
We obtain an elementary invariance principle for multi-dimensional Brownian sheet where the underlying random fields are not necessarily independent or stationary. Possible applications include unit-root tests for spatial as well as panel data models.
In this paper, we obtain sufficient conditions in terms of projective criteria under which the partial sums of a stationary process with values in ${\mathcal{H}}$ (a real and separable Hilbert space) admits an approximation, in ${\mathbb{L}}^p({\mathcal{H}})$, $p>1$, by a martingale with stationary differences, and we then estimate the error of approximation in ${\mathbb{L}}^p({\mathcal{H}})$. The results are exploited to further investigate the behavior of the partial sums. In particular we obtain new projective conditions concerning the Marcinkiewicz-Zygmund theorem, the moderate deviations …
We prove the almost sure invariance principle for stationary R^d--valued processes (with dimension-independent very precise error terms), solely under a strong assumption on the characteristic functions of these processes. This assumption is easy to check for large classes of dynamical systems or Markov chains, using strong or weak spectral perturbation arguments.
We consider "randomized" statistics constructed by using a finite number of observations a random field at randomly chosen points. We generalize the invariance principle (the functional CLT), the Glivenko--Cantelli theorem, the theorem about convergence to the Brownian bridge and the Kolmogorov theorem about the limit distribution of the empirical distribution function, as well as an improved version of the CLT in A. Tempelman, Randomized multivariate central limit theorems for ergodic homogeneous random fields, Stochastic Processes and their Applications. 143 (2022), 89-105. The randomized ap…
We consider the random conductance model in a stationary and ergodic environment. Under suitable moment conditions on the conductances and their inverse, we prove a quenched invariance principle for the random walk among the random conductances. The moment conditions improve earlier results of Andres, Deuschel and Slowik [Ann.\ Probab.] and are the minimal requirement to ensure that the corrector is sublinear everywhere. The key ingredient is an essentially optimal deterministic local boundedness result for finite difference equations in divergence form.
In this dissertation, we show that the Central Limit Theorem and the Invariance Principle for Discrete Fourier Transforms discovered by Peligrad and Wu can be extended to the quenched setting. We show that the random normalization introduced to extend these results is necessary and we discuss its meaning. We also show the validity of the quenched Invariance Principle for fixed frequencies under some conditions of weak dependence. In particular, we show that this result holds in the martingale case. The discussion needed for the proofs allows us to show some general facts apparently not noticed…
We extend the spectral approach of S. Gouëzel for the vector-valued almost sure invariance principle (ASIP) to certain classes of non-stationary sequences with a weaker control over the behavior of the covariance matrices, assuming only linear growth. Then we apply this extension to obtain a quenched vector-valued ASIP for random perturbations of a fixed Anosov diffeomorphism as well as random perturbations of a billiard map associated to the periodic Lorentz gas. We also consider certain classes of random piecewise expanding maps.
Understanding the statistical properties of the aperiodic planar Lorentz gas stands as a grand challenge in the theory of dynamical systems. Here we study a greatly simplified but related model, proposed by Arvind Ayyer and popularized by Joel Lebowitz, in which a scatterer configuration on the torus is randomly updated between collisions. Taking advantage of recent progress in the theory of time-dependent billiards on the one hand and in probability theory on the other, we prove a vector-valued almost sure invariance principle for the model. Notably, the configuration sequence can be weakly d…
In this paper, we give explicit rates in the central limit theorem and in the almost sure invariance principle for general R d-valued cocycles that appear in the study of the left random walk on linear groups. Our method of proof lies on a suitable martingale approximation and on a careful estimation of some coupling coefficients linked with the underlying Markov structure. Concerning the martingale part, the available results in the literature are not accurate enough to give almost optimal rates whether in the central limit theorem for the Wasserstein distance, or in the strong approximation.…
In this paper, we obtain precise rates of convergence in the strong invariance principle for stationary sequences of real-valued random variables satisfying weak dependence conditions including strong mixing in the sense of Rosenblatt (1956) as a special case. Applications to unbounded functions of intermittent maps are given.
We prove that, for (adapted) stationary processes, the so-called Maxwell-Wood-roofe condition is sufficient for the law of the iterated logarithm and that it is optimal in some sense. We obtain a similar conclusion concerning the Marcinkiewicz-zygmund strong law of large numbers. Those results actually hold in the context of Banach valued stationary processes, including the case of $L^r$-valued random variables, with $1\le r<\infty$. In this setting we also prove the weak invariance principle, under a version of the Maxwell-Woodroofe condition, generalizing a result of Peligrad and Utev \cite{…
There has been recent interest in the conditional central limit question for (strictly) stationary, ergodic processes $...,X_{-1},X_0,X_1,...$ whose partial sums $S_n=X_1+...+X_n$ are of the form $S_n=M_n+R_n$, where $M_n$ is a square integrable martingale with stationary increments and $R_n$ is a remainder term for which $E(R_n^2)=o(n)$. Here we explore the law of the iterated logarithm (LIL) for the same class of processes. Letting $\Vert\cdot\Vert$ denote the norm in $L^2(P)$, a sufficient condition for the partial sums of a stationary process to have the form $S_n=M_n+R_n$ is that $n^{-3/2…
We consider a stochastic billiard in a random tube which stretches to infinity in the direction of the first coordinate. This random tube is stationary and ergodic, and also it is supposed to be in some sense well behaved. The stochastic billiard can be described as follows: when strictly inside the tube, the particle moves straight with constant speed. Upon hitting the boundary, it is reflected randomly, according to the cosine law: the density of the outgoing direction is proportional to the cosine of the angle between this direction and the normal vector. We also consider the discrete-time …
Invariance principles for rough walks in random conductances
University of California, Los Angeles（加州大学洛杉矶分校）；Technische Universität München（慕尼黑工业大学）；Università degli Studi di Milano-Bicocca（米兰比可卡大学）；University of Mannheim（曼海姆大学）
We consider a renewal process with regularly varying stationary and weakly dependent steps, and prove that the steps made before a given time $t$, satisfy an interesting invariance principle. Namely, together with the age of the renewal process at time $t$, they converge after scaling to the Pitman--Yor distribution. We further discuss how our results extend the classical Dynkin--Lamperti theorem.
We consider a stationary and ergodic random field $\{ω(e) : e \in E_d\}$ that is parameterized by the edge set of the Euclidean lattice $\mathbb{Z}^d$, $d \geq 2$. The random variable $ω(e)$, taking values in $[0, \infty)$ and satisfying certain moment bounds, is thought of as the conductance of the edge $e$. Assuming that the set of edges with positive conductances give rise to a unique infinite cluster $\mathcal{C}_{\infty}(ω)$, we prove a quenched invariance principle for the continuous-time random walk among random conductances under relatively mild conditions on the structure of the infin…
We prove an invariance principle for a class of tilted (1+1)-dimensional SOS models or, equivalently, for a class of tilted random walk bridges in Z_+. The limiting objects are stationary reversible ergodic diffusions with drifts given by the logarithmic derivatives of the ground states of associated singular Sturm-Liouville operators. In the case of a linear area tilt, we recover the Ferrari-Spohn diffusion with log-Airy drift, which was derived by Ferrari and Spohn in the context of Brownian motions conditioned to stay above circular and parabolic barriers.
This paper establishes a central limit theorem and an invariance principle for a wide class of stationary random fields under natural and easily verifiable conditions. More precisely, we deal with random fields of the form $X_k = g(\varepsilon_{k-s}, s \in \Z^d)$, $k\in\Z^d$, where $(\varepsilon_i)_{i\in\Z^d}$ are i.i.d random variables and $g$ is a measurable function. Such kind of spatial processes provides a general framework for stationary ergodic random fields. Under a short-range dependence condition, we show that the central limit theorem holds without any assumption on the underlying d…
