Extensions of results of Komlós, Major, and Tusnády to the multivariate case

A construction method is developed which enables us to establish the well-known approximation results of Komlós, Major, Tusnády (Z. Wahrsch. Verw. Gebiete34 33-58) in the multidimensional setting. The basic tool is a large deviation theorem for conditional distribution functions which may be of independent interest.