Weak Invariance Principle Research Articles

On the weak invariance principle for ortho-martingale in Banach spaces. Application to stationary random fields

In this paper, we study the weak invariance principle for stationary ortho-martingales with values in 2-smooth or cotype 2 Banach spaces. Then, with the help of a suitable maximal ortho-martingale approximation, we derive the weak invariance principle for stationary random fields in Lp, 1≤p≤2, under a condition in the spirit of Hannan. As an application, we get an asymptotic result for the Lp-distances (1≤p≤2) between the common distribution function and the corresponding empirical distribution function of stationary random fields.

Statistical inference for the slope parameter in functional linear regression

In this paper, we consider the linear regression model Y=SX+ε with functional regressors and responses. We develop new inference tools to quantify deviations of the true slope S from a hypothesized operator S0 with respect to the Hilbert–Schmidt norm ~S−S0~2, as well as the prediction error E‖SX−S0X‖2. Our analysis is applicable to functional time series and based on asymptotically pivotal statistics. This makes it particularly user-friendly, because it avoids the choice of tuning parameters inherent in long-run variance estimation or bootstrap of dependent data. We also discuss two sample problems as well as change point detection. Finite sample properties are investigated by means of a simulation study. Mathematically, our approach is based on a sequential version of the popular spectral cut-off estimator SˆN for S. We prove that (sequential) plug-in estimators of the deviation measures are N-consistent and satisfy weak invariance principles. These results rest on the smoothing effect of L2-norms, that we exploit by a new proof-technique, the smoothness shift, which has potential applications in other fields.

On the weak invariance principle for non-adapted stationary random fields under projective criteria

In this paper, we study the central limit theorem (CLT) and its weak invariance principle (WIP) for sums of stationary random fields non-necessarily adapted, under different normalizations. To do so, we first state sufficient conditions for the validity of a suitable ortho-martingale approximation. Then, with the help of this approximation, we derive projective criteria under which the CLT as well as the WIP hold. These projective criteria are in the spirit of Hannan’s condition and are well adapted to linear random fields with ortho-martingale innovations and which exhibit long memory.

Mixing rates for potentials of non-summable variations

AbstractMixing rates, relaxation rates, and decay of correlations for dynamics defined by potentials with summable variations are well understood, but little is known for non-summable variations. This paper exhibits upper bounds for these quantities for dynamics defined by potentials with square-summable variations. We obtain these bounds as corollaries of a new block coupling inequality between pairs of dynamics starting with different histories. As applications of our results, we prove a new weak invariance principle and a Hoeffding-type inequality.

On the dimension of the space of harmonic functions on transitive shift spaces

In this paper, we show a new relation between phase transition in Statistical Mechanics and the dimension of the space of harmonic functions (SHF) for a transfer operator. This is accomplished by extending the classical Ruelle-Perron-Frobenius theory to the realm of low regular potentials defined on either finite or infinite (uncountable) alphabets. We also give an example of a potential having a phase transition where the Perron-Frobenius eigenvector space has dimension two. We discuss entropy and equilibrium states, in this general setting, and show that if the SHF is non-trivial, then the associated equilibrium states have full support. We also obtain a weak invariance principle in cases where the spectral gap property is absent. As a consequence, a functional central limit theorem for non-local observables of the Dyson model is obtained.

Changepoint in Error-Prone Relations

Linear relations, containing measurement errors in input and output data, are considered. Parameters of these so-called errors-in-variables models can change at some unknown moment. The aim is to test whether such an unknown change has occurred or not. For instance, detecting a change in trend for a randomly spaced time series is a special case of the investigated framework. The designed changepoint tests are shown to be consistent and involve neither nuisance parameters nor tuning constants, which makes the testing procedures effortlessly applicable. A changepoint estimator is also introduced and its consistency is proved. A boundary issue is avoided, meaning that the changepoint can be detected when being close to the extremities of the observation regime. As a theoretical basis for the developed methods, a weak invariance principle for the smallest singular value of the data matrix is provided, assuming weakly dependent and non-stationary errors. The results are presented in a simulation study, which demonstrates computational efficiency of the techniques. The completely data-driven tests are illustrated through problems coming from calibration and insurance; however, the methodology can be applied to other areas such as clinical measurements, dietary assessment, computational psychometrics, or environmental toxicology as manifested in the paper.

Some limit theorems for dependent Bernoulli random variables

We consider a sequence of correlated Bernoulli variables whose probability of success of the current trial depends conditionally on the previous trials as a linear function of the sample mean. We extend the results of Zhang and Zhang (2015) by establishing an almost sure invariance principle and a weak invariance principle in a larger setting. Moreover, we also state a Gaussian fluctuation related to an almost sure and Lp convergence for the model.

Multidimensional multiscale scanning in exponential families: Limit theory and statistical consequences

We consider the problem of finding anomalies in a $d$-dimensional field of independent random variables $\{Y_{i}\}_{i\in \{1,\ldots,n\}^{d}}$, each distributed according to a one-dimensional natural exponential family $\mathcal{F}=\{F_{\theta }\}_{\theta \in \Theta }$. Given some baseline parameter $\theta _{0}\in \Theta $, the field is scanned using local likelihood ratio tests to detect from a (large) given system of regions $\mathcal{R}$ those regions $R\subset \{1,\ldots,n\}^{d}$ with $\theta _{i}\neq \theta _{0}$ for some $i\in R$. We provide a unified methodology which controls the overall familywise error (FWER) to make a wrong detection at a given error rate. Fundamental to our method is a Gaussian approximation of the distribution of the underlying multiscale test statistic with explicit rate of convergence. From this, we obtain a weak limit theorem which can be seen as a generalized weak invariance principle to nonidentically distributed data and is of independent interest. Furthermore, we give an asymptotic expansion of the procedures power, which yields minimax optimality in case of Gaussian observations.

Uniform asymptotic normality of self-normalized weighted sums of random variables*

Let X, X1, X2, . . . be a sequence of nondegenerate i.i.d. random variables, let μ = {μni : n ∈ ℕ+, i = 1, …, n} be a triangular array of possibly random probabilities on the interval [0, 1], and let $$ \mathcal{F} $$ be a class of functions with bounded q-variation on [0, 1] for some q ∈ [1, 2). We prove the asymptotic normality uniformly over $$ \mathcal{F} $$ of self-normalized weighted sums $$ {\sum}_{i=1}^n{X}_i{\mu}_{ni} $$ when μ is the array of point measures, uniform probabilities, and their random versions. Also, we prove a weak invariance principle in the Banach space of functions of bounded p-variation with p > 2 for partial-sum processes, polygonal processes, and their adaptive versions.

Rate of Convergence in the Weak Invariance Principle for Deterministic Systems

We obtain the first results on convergence rates in the Prokhorov metric for the weak invariance principle (functional central limit theorem) for deterministic dynamical systems. Our results hold for uniformly expanding/hyperbolic (Axiom A) systems, as well as nonuniformly expanding/hyperbolic systems such as dispersing billiards, Hénon-like attractors, Viana maps and intermittent maps. As an application, we obtain convergence rates for deterministic homogenization in multiscale systems.

On limit theorems for fields of martingale differences

We prove a central limit theorem for stationary multiple (random) fields of martingale differences f∘Ti̲, i̲∈Zd, where Ti̲ is a Zd action. In most cases the multiple (random) fields of martingale differences is given by a completely commuting filtration. A central limit theorem proving convergence to a normal law has been known for Bernoulli random fields and in Volný (2015) this result was extended to random fields where one of generating transformations is ergodic.In the present paper it is proved that a convergence takes place always and the limit law is a mixture of normal laws. If the Zd action is ergodic and d≥2, the limit law need not be normal.For proving the result mentioned above, a generalisation of McLeish’s CLT for arrays (Xn,i) of martingale differences is used. More precisely, sufficient conditions for a CLT are found in the case when the sums ∑iXn,i2 converge only in distribution.The CLT is followed by a weak invariance principle. It is shown that central limit theorems and invariance principles using martingale approximation remain valid in the non-ergodic case.

Hölderian weak invariance principle under the Maxwell and Woodroofe condition

We investigate the weak invariance principle in H{\"o}lder spaces under some reinforcement of the Maxwell and Woodroofe condition.

Martingale-coboundary representation for stationary random fields

We prove a martingale-coboundary representation for random fields with a completely commuting filtration. For random variables in [Formula: see text], we present a necessary and sufficient condition which is a generalization of Heyde’s condition for one-dimensional processes from 1975. For [Formula: see text] spaces with [Formula: see text] we give a necessary and sufficient condition which extends Volný’s result from 1993 to random fields and improves condition of El Machkouri and Giraudo from 2016. A new sufficient condition is presented which for dimension one improves Gordin’s condition from 1969. In application, new weak invariance principle and estimates of large deviations are found.

Weak invariance principle in some Besov spaces for stationary martingale differences

We extend the classical Donsker weak invariance principle to some Besov space framework. We consider polygonal line processes built from partial sums of stationary martingale differences and of independent and identically distributed random variables. The results obtained are shown to be optimal.

Relative complexity of random walks in random scenery in the absence of a weak invariance principle for the local times

We answer a question of Aaronson about the relative complexity of Random Walks in Random Sceneries driven by either aperiodic two-dimensional random walks, two-dimensional Simple Random walk, or by aperiodic random walks in the domain of attraction of the Cauchy distribution. A key step is proving that the range of the random walk satisfies the Fölner property almost surely.

Invariance principles under the Maxwell–Woodroofe condition in Banach spaces

We prove that, for (adapted) stationary processes, the so-called Maxwell–Woodroofe condition is sufficient for the law of the iterated logarithm and that it is optimal in some sense. That result actually holds in the context of Banach valued stationary processes, including the case of $L^{p}$-valued random variables, with $1\le p<\infty$. In this setting, we also prove the weak invariance principle, hence generalizing a result of Peligrad and Utev [Ann. Probab. 33 (2005) 798–815]. The proofs make use of a new maximal inequality and of approximation by martingales, for which some of our results are also new.

QUENCHED CENTRAL LIMIT THEOREMS FOR A STATIONARY LINEAR PROCESS

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.

Deviation inequalities for martingales with applications

Using changes of probability measure developed by Grama and Haeusler (2000) [18], we extend the deviation inequalities of Lanzinger and Stadtmüller (2000) [26] and Fuk and Nagaev (1971) [15] to the case of martingales. Our inequalities recover the best possible decaying rate in the independent case. In particular, these inequalities improve the results of Lesigne and Volný (2001) [27] under a stronger condition that the martingale differences have bounded conditional moments. Applications to linear regressions with martingale difference innovations, weak invariance principles for martingales and self-normalized deviations are provided. In particular, we establish a type of self-normalized deviation bounds for parameter estimation of linear regressions. Such type bounds have the advantage that they do not depend on the distribution of the regression random variables.

Limit theorems for dependent Bernoulli variables with statistical inference

ABSTRACTWe consider a class of dependent Bernoulli variables where the conditional success probability is a linear combination of the last few trials and the original success probability. We obtain its limit theorems including the strong law of large numbers, weak invariance principle, and law of the iterated logarithm. We also derive some statistical inference results which make the model applicable. Simulation results are exhibited as well to show that with small sample size the convergence rate is satisfying and the proposed estimators behave well.

A note on statistical properties for nonuniformly hyperbolic systems with slow contraction and expansion

We provide a systematic approach for deducing statistical limit laws via martingale-coboundary decomposition, for nonuniformly hyperbolic systems with slowly contracting and expanding directions. In particular, if the associated return time function is square-integrable, then we obtain the central limit theorem, the weak invariance principle, and an iterated version of the weak invariance principle.