Strong invariance principles for partial sums of independent random vectors
type: Item
instance of: scholarly article
title: Strong invariance principles for partial sums of independent random vectors
zbMATH Open document ID: 0637.60041
author: Uwe Einmahl
published in: The Annals of Probability
publication date: 1987-01-01T00:00:00Z
review text: Let Q be a Borel probability measure on the Euclidean space \(R^ d\) with mean zero and covariance matrix \(\Sigma\) and let H be a continuous nonnegative function on [0,\(\infty)\) satisfying some regularity conditions. It is shown that there exist sequences of independent random vectors \(\{X_ n\}\), \(\{Y_ n\}\) such that \(X_ n\) has distribution Q, \(Y_ n\) has distribution \(N(0,\Sigma_ n)\), \(\| \Sigma_ n-\Sigma \| =o((H^{-1}(n))^ 2/n)\) and \((X_ 1+...+X_ n)-(Y_ 1+...+Y_ n)=o(H^{-1}(n))\) a.s. Here \(\| \cdot \|\) denotes the Euclidean matrix norm and \(H^{-1}\) the inverse function of H. This result immediately implies the multidimensional version of Strassen's invariance principle, \textit{P. Major}'s strong invariance principle [Z. Wahrscheinlichkeitstheor. verw. Geb. 35, 213-220 (1976; Zbl 0338.60031)] under weakened assumptions as well as a new strong invariance principle which ``interpolates'' between these two invariance principles. It is shown that the assumptions are nearly necessary. The main tools in the proof are the Strassen-Dudley theorem and a new estimate of the Prokhorov distance in the multidimensional central limit theorem which is a modification of a well-known theorem of \textit{V. V. Yurinskij} [Teor. Verojatn. Primen. 20, 3-12 (1975; Zbl 0351.60007); English translation in Theor. Probab. Appl. 20, 1-10 (1975)].
Mathematics Subject Classification ID: 60F17
Mathematics Subject Classification ID: 60F15
Mathematics Subject Classification ID: 60G50
zbMATH DE Number: 4038917
zbMATH Keywords: strong invariance principle
zbMATH Keywords: Prokhorov distance
zbMATH Keywords: multidimensional version of Strassen's invariance principle
zbMATH Keywords: multidimensional central limit theorem
reviewed by: T. Inglot
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aop/1176991985
OpenAlex ID: W2089808030
DOI: 10.1214/AOP/1176991985
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description: scientific article; zbMATH DE number 4038917