The invariance principle for triangular arrays of dependent variables is studied. We use the concept of mixingale, proposed by McLeish (1975). Uniform bounds are imposed on the growth of conditional expectations with respect to distant predecessors. The theorem is applied to invariance principles for autocovariance estimates based on triangular arrays of time-series data for weak mixing sequences and linear processes. Such results are required for bootstrap applications.



Les auteurs etudient le principe d'invariance pour des tableaux triangulaires de variables dependantes. Us utilisent pour ce faire le concept de 'mixingale' propose par McLeish (1975). Us imposent des bornes uniformes sur la croissance des esperances conditionnelles par rapport aux predecesseurs eloignes. En decoulent des applications aux principes d'invariance pour des estimations de I'autocovariance s'appuyant sur des tableaux triangulaires de series chronologiques pour des suites faiblement melangeantes et des processus lineaires. De tels resultats sont prerequis pour la mise en oeuvre de methodes d'auto-amorcage.

An invariance principle for triangular arrays of dependent variables with application to autocovariance estimation

Andre Robert Dabrowski, Professor of Mathematics and Dean of the Faculty of Sciences at the University of Ottawa, died October 7, 2006, after a short battle with cancer The author of the present paper, a long-term friend and collaborator of Andre Dabrowski, gives a survey of Andre's work on weak dependence and limit theorems in probability theory The Canadian Journal of Statistics 37: 307–326; 2009 © 2009 Statistical Society of Canada

https://www.ruhr-uni-bochum.de/imperia/md/content/mathematik/lehrstuhlxii/ard_memorial.pdf

André Dabrowski's work on limit theorems and weak dependence

In this paper, the authors obtain the strong approximations, the laws of the single logarithm, the functional laws of the single logarithm, and the limit law of the single logarithm for linear processes generated by arrays of independent and identically distributed random variables, and they show that they are all equivalent under the same moment conditions.

/pdf/strong-law-for-linear-processes-35bl2tvzu8.pdf

Strong law for linear processes

We establish a strong invariance principle for triangular arrays of a broad class of weakly dependent real random variables. We approximate the original array of dependent random variables by an array of rowwise independent standard normal variables. We demonstrate the functional central limit theorem and law of the iterated logarithm for the approximating array and thereby extend these results to the original array. Among several examples, we look at arrays used in describing the rate of convergence of estimators in regression analysis.



Nous etablissons un principe d'invariance fort pour des tableaux triangulaires de variables aleatoires faiblement dependentes. Le tableau donne est approximise par un tableau de variables aleatoires normales standards et independentes par rangee. On demontre les formes fonctionnelles de la loi de la limite centrale et du logarithme itere pour le tableau des normales, et donc pour le tableau initial. Parmi quelques examples, on regarde des tableaux utilises dans l'etude du taux de convergence des estimateurs en analyse de la regression.

Abdelhak Zoglat

Papers

Strong invariance principles for triangular arrays of weakly dependent random variables