Multidimensional variant of the Komlós, Major and Tusnády results for vectors with finite exponential moments (Q1595733)

The author announces a new strong invariance principle for independent and non-identically distributed random vectors which essentially improves the result obtained by \textit{U. Einmahl} [J. Multivariate Anal. 28, No. 1, 20-68 (1989; Zbl 0676.60038)]. The main theorem is as follows: Let \(\tau\geq 1\) and \(\alpha>0\) and \(\xi_1,\dots,\xi_n\) be \(d\)-dimensional random vectors with mean 0, covariances being the identity matrix and with distributions having finite exponential moments. Then there exists a probability space and independent random vectors \(X_1,\dots,X_n\) and independent Gaussian random vectors \(Y_1,\dots,Y_n\) on it such that \(\xi_k\overset\text{D}=X_k\) for \(k=1,\dots,n\) and \[ E\exp\bigl\{c_1 \tau^{-1}d^{-3} (\log^+d)^{-1}\Delta (X,Y) \bigr\}\leq\exp \bigl\{c _2d^{9/4+ \alpha} \log^+ (n/\tau^2)\bigr\}, \] where \(\Delta(X,Y)= \sup_{1\leq k\leq n}\max_{1\leq j\leq d}|\sum^k_{i=1} (X_{ij}-Y_{ij}) |\) and \(c_1,c_2\) depend only on \(\alpha\). -- Contrary to Einmahl's paper explicit dependence of constants on the dimension \(d\) is determined.