The Annals of Probability
- Ann. Probab.
- Volume 15, Number 4 (1987), 1419-1440.
Strong Invariance Principles for Partial Sums of Independent Random Vectors
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.
Ann. Probab., Volume 15, Number 4 (1987), 1419-1440.
First available in Project Euclid: 19 April 2007
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 60F17: Functional limit theorems; invariance principles
Secondary: 60F15: Strong theorems
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