Open Access
October, 1987 Strong Invariance Principles for Partial Sums of Independent Random Vectors
Uwe Einmahl
Ann. Probab. 15(4): 1419-1440 (October, 1987). DOI: 10.1214/aop/1176991985


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.


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Uwe Einmahl. "Strong Invariance Principles for Partial Sums of Independent Random Vectors." Ann. Probab. 15 (4) 1419 - 1440, October, 1987.


Published: October, 1987
First available in Project Euclid: 19 April 2007

zbMATH: 0637.60041
MathSciNet: MR905340
Digital Object Identifier: 10.1214/aop/1176991985

Primary: 60F17
Secondary: 60F15

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

Rights: Copyright © 1987 Institute of Mathematical Statistics

Vol.15 • No. 4 • October, 1987
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