ESAIM Probability and Statistics April Vol MULTIDIMENSIONAL VERSION OF THE RESULTS OF KOML OS MAJOR AND TUSN ADY FOR VECTORS WITH FINITE EXPONENTIAL MOMENTS

A multidimensional version of the results of Koml os Ma jor and Tusn ady for the Gaussian approximation of the sequence of successive sums of independent random vectors with nite exponential moments is obtained Introduction The statement of the problem is well known It is required to construct on a probability space a sequence of independent random vectors X Xn with given distributions and a corresponding sequence of independent Gaussian random vectors Y Yn this means that Yi has the same mean and the same covariance operator as Xi i n so that the quantity X Y max k n k P i Xi k P i Yi would be so small as possible with large probability Here jxj max jxj j for x x xd R Sometimes we shall use another notation for sum mands and by analogy with notation in Sakhanenko write e X e Y max k n k P i e Xi k P i e Yi and so on In some sense this problem is one of the most important in probability approximations because many well known probability theorems can be considered as consequences of results about strong approximation of sequences of sums by corresponding Gaussian se quences This is related to the law of iterated logarithm to several theo rems about large deviations to the estimates for the rate of convergence of the Prokhorov distance in the invariance principles see Prokhorov Skorokhod Borovkov as well as to the Strassen type approximations see for example Cs org o and Hall The rate of approximation in the invariance principle was studied by many authors see e g Prokhorov Skorokhod Borovkov Cs org o and R ev esz and the bibliography in Cs org o and R ev esz URL address of the journal http www emath fr ps Research supported by the Alexander von Humboldt Foundation by the Russian Foun dation of Basic Research grants and by grant RFBR DFG ge and by the International Science Foundation grant R Received by the journal April Revised November Accepted for publication January c Soci et e de Math ematiques Appliqu ees et Industrielles Typeset by TEX