The Annals of Probability

Strong Invariance Principles for Partial Sums of Independent Random Vectors

Uwe Einmahl

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Abstract

An estimate in the multidimensional central limit theorem is obtained, which is used together with the Strassen-Dudley theorem to prove a strong approximation theorem for partial sums of independent, identically distributed $d$-dimensional random vectors. This theorem implies immediately multi-dimensional versions of the strong invariance principles of Strassen and Major as well as a new $d$-dimensional strong invariance principle which improves the known results for the 1-dimensional case. In particular, we are able to weaken the assumption in Major's strong invariance principle. At the same time, it is shown that the assumptions of our theorem are nearly necessary.

Article information

Source
Ann. Probab., Volume 15, Number 4 (1987), 1419-1440.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176991985

Digital Object Identifier
doi:10.1214/aop/1176991985

Mathematical Reviews number (MathSciNet)
MR905340

Zentralblatt MATH identifier
0637.60041

JSTOR
links.jstor.org

Subjects
Primary: 60F17: Functional limit theorems; invariance principles
Secondary: 60F15: Strong theorems

Keywords
Strong invariance principles partial sums of independent random vectors strong approximations

Citation

Einmahl, Uwe. Strong Invariance Principles for Partial Sums of Independent Random Vectors. Ann. Probab. 15 (1987), no. 4, 1419--1440. doi:10.1214/aop/1176991985. https://projecteuclid.org/euclid.aop/1176991985


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