Weak Invariance Principle Research Articles

Donsker's theorem for self-normalized partial sums processes

Let $X, X_1, X_2,\ldots$ be a sequence of nondegenerate i.i.d. random variables with zero means. In this paper we show that a self-normalized version of Donsker's theorem holds only under the assumption that X belongs to the domain of attraction of the normal law. A thus resulting extension of the arc sine law is also discussed. We also establish that a weak invariance principle holds true for self-normalized, self-randomized partial sums processes of independent random variables that are assumed to be symmetric around mean zero, if and only if $\max_{1\le j\le n}|X_j|/V_n\to_P 0$, as $n\to\infty$, where $V_n^2=\sum_{j=1}^{n}X_j^2$.

Entropy fluctuations for parabolic maps

We prove log-normal fluctuations and the weak-invariance principle for the convergence to the entropy in the Ornstein–Weiss theorem for a class of parabolic maps of the interval. For such maps, we also compute the Lyapunov exponent using the linear recurrence of the returns of cylinders into themselves.

Random iteration of Euclidean isometries

We consider the statistical behaviour of independent identically distributed compositions of a finite set of Euclidean isometries of ℝn. We give a new proof of the central limit theorem and weak invariance principles, and we obtain the law of the iterated logarithm. Our results generalize immediately to Markov chains.We also give simple geometric criteria for orbits to grow linearly or sublinearly with probability one and for nondegeneracy (nonsingular covariance matrix) in the statistical limit theorems.Our proofs are based on dynamical systems theory rather than a purely probabilistic approach.

Non-autonomous systems: asymptotic behaviour and weak invariance principles

Results pertaining to asymptotic behaviour of solutions of non-autonomous ordinary differential equations with locally integrably bounded right-hand sides are presented. Ramifications for weakly asymptotically autonomous systems and adaptively controlled systems are highlighted.

On the Central Limit Theorem and Its Weak Invariance Principle for Strongly Mixing Sequences with Values in a Hilbert Space via Martingale Approximation

In this paper we not only prove an extension to Hilbert spaces of a sharp central limit theorem for strongly real-valued mixing sequences, but also slightly improve it. The proof is mainly based on the Bernstein blocking technique and approximations by martingale differences. Moreover, we derive also the corresponding functional central limit theorem.

Invariance Principles for Interval Maps with an Indifferent Fixed Point

In this note we establish an almost sure invariance principle for a large class of interval maps with indifferent fixed points, including the Manneville-Pomeau map. This implies a number of well-known corollaries, including the Weak Invariance Principle and the Law of the Iterated Logarithm.

A note on estimating the change-point of a gradually changing stochastic process

We consider an estimator of the change-point of a stochastic process satisfying some weak invariance principles. Making use of the known asymptotics of the approximating Wiener processes we derive various limiting distributions according to different orders of magnitude of the underlying change. The results take into account, but also extend those of Hušková (J. Statist. Plann. Infer. 76 (1999) 109–125), who studied a location model for gradual changes with independent, identically distributed (iid) errors. Aim of this note is to show that corresponding results hold also true in our more general setting.

On impulsive processes

Consider impulsive processes that consists of between two impulses non-autonomous, non-linear continuous processes subjected to time-varying impulsive constraints. By extending compact processes defined by Dafermos for non-autonomous systems without impulses, a class of regular impulsive processes is provided and its applications are stressed by selected examples. A weak invariance principle and an invariance principle are established for the regular impulsive processes. The paper thus lays ground work for geometric theory of general impulsive processes.

An estimator of the number of change points based on a weak invariance principle

We study an estimator of the number of change points in the drift of a stochastic process based on the Schwarz criterion. In a general statistical model where the additive measurement noise satisfies a certain weak invariance principle (examples included are partial sums, renewal processes, and linear processes in time series analysis) consistency can be shown under the condition that the number of jumps is not greater than a given upper bound.

Testing for changes in the mean or variance of a stochastic process under weak invariance

Asymptotic CUSUM tests are derived for detecting changes in the mean or variance of a stochastic process for which a weak invariance principle is available. Conditions for the consistency of these tests are also discussed.

Invariance principles and Gaussian approximation for strictly stationary processes

We show that in any aperiodic and ergodic dynamical system there exists a square integrable process ( f ∘ T i ) (f\circ T^{i}) the partial sums of which can be closely approximated by the partial sums of Gaussian i.i.d. random variables. For ( f ∘ T i ) (f\circ T^{i}) both weak and strong invariance principles hold.

Nonparametric testing for correlation models with dependent data

We define a family of nonparametric goodness-of-fit tests for the regression function in correlation and autoregression models with strongly mixing observations. These tests are based on the limiting functional distribution of a certain sequence of processes. These processes turn out to converge, under the null hypothesis H 0, to a tractable time-changed Wiener process. This result is deduced from our main theorem (Theorem 1) which establishes a weak invariance principle for this sequence of processes. We focus then on two especially interesting tests within the family under consideration. The first one is analogous to the Kolmogorov-Smirnov type test and the second one is analogous to the Cramér-von Mises type test. Using our preceding results, we derive estimates for the asymptotic behavior of the power of the Kolmogorov-Smirnov type test against both fixed and local alternatives. Simulation results giving information on how good the approximation by the asymptotics is and on the power of the test are provided. Comparisons are made with other tests.

Parallel and bootstrapped stochastic approximation

Parallelization of stochastic approximation procedures can reduce computation and total observation time of a system. Concerning the number of all observations used by the pure sequential and the suggested parallel method a weak invariance principle implies the asymptotic equivalence of both methods. A loglog invariance principle and a rate of a.s. convergence result describe the pathwise properties. Due to the parallel design asymptotic confidence regions can readily be constructed either by computing the bootstrap distribution or the Gaussian limit distribution determined by the empirical covariance

A Non‐parametric Test for Generalized First‐order Autoregressive Models

We derive a non‐parametric test for testing the presence of V(Xi,εi) in the non‐parametric first‐order autoregressive model Xi+1=T(Xi)+V(Xi,εi)+U(Xi)εi+1, where the function T(x) is assumed known. The test is constructed as a functional of a basic process for which we establish a weak invariance principle, under the null hypothesis and under stationarity and mixing assumptions. Bounds for the local and non‐local powers are provided under a condition which ensures that the power tends to one as the sample size tends to infinity.The testing procedure can be applied, e.g. to bilinear models, ARCH models, EXPAR models and to some other uncommon models. Our results confirm the robustness of the test constructed in Ngatchou Wandji (1995) and in Diebolt & Ngatchou Wandji (1995).

A note on the weak invariance principle for local times

We prove uniform L 2-convergence of local times of aperiodic recurrent random walks to local times of strictly α-stable processes with α > 1.

Invariance principles for deconvoluting kernel density estimation

A method of obtaining weak or strong invariance principles for the centered and normalized deconvoluting kernel estimator is presented. It is shown that these approximations, under certain smoothness constraints, achieve the best global rates of convergence.

Weak invariance principle for integral type functionals

Weak invariance principle for integral type functionals

Strong approximation for cross-covariances of linear variables with long-range dependence

Suppose { ε k , −∞ < k < ∞} is an independent, not necessarily identically distributed sequence of random variables, and { c j } ∞ j=0 , { d j } ∞ j=0 are sequences of real numbers such that Σ j c 2 j < ∞, Σ j d 2 j < ∞. Then, under appropriate moment conditions on { ε k , −∞ < k < ∞}, y k ≜Σ ∞ j=0c jε k-j , z k ≜ Σ ∞ j=0d jε k-j exist almost surely and in L 4 and the question of Gaussian approximation to S [t]≜Σ [t] k=1 (y k z k − E{y k z k}) becomes of interest. Prior to this work several related central limit theorems and a weak invariance principle were proven under stationary assumptions. In this note, we demonstrate that an almost sure invariance principle for S [ t] , with error bound sharp enough to imply a weak invariance principle, a functional law of the iterated logarithm, and even upper and lower class results, also exists. Moreover, we remove virtually all constraints on ε k for “time” k ≤ 0, weaken the stationarity assumptions on { ε k , −∞ < k < ∞}, and improve the summability conditions on { c j } ∞ j=0 , { d j } ∞ j=0 as compared to the existing weak invariance principle. Applications relevant to this work include normal approximation and almost sure fluctuation results in sample covariances (let d j = c j- m for j ≥ m and otherwise 0), quadratic forms, Whittle's and Hosoya's estimates, adaptive filtering and stochastic approximation.

Limit theorems and Markov approximations for chaotic dynamical systems

We prove the central limit theorem and weak invariance principle for abstract dynamical systems based on bounds on their mixing coefficients. We also develop techniques of Markov approximations for dynamical systems. We apply our results to expanding interval maps, Axiom A diffeomorphisms, chaotic billiards and hyperbolic attractors.

Central Limit Theorem, Weak Law of Large Numbers for Martingales in Banach Spaces, and Weak Invariance Principle - A Quantitative Study

This article deals with quantitative results by involving the standard modulus of continuity in Banach spaces. These concern convergence in distribution for Banach space-valued martingale difference sequences and weak convergence of the distribution of random polygonal lines to the Wiener-measure on C([0, 1]). A general theorem is given with applications to the central limit theorem and weak law of large numbers for Banach space-valued martingales. Another general theorem is presented on the weak invariance principle with an application to a central limit theorem for real-valued martingales. The exposed results generalize earlier related results of Butzer, Hahn, Kirschfink, and Roeckerath.