Multidimensional version of the results of Komlos, Major and Tusnady for vectors with finite exponential moments

Abstract

Abstract. A m ultidimensional version of the results of Koml os, Major and Tusn ady for the Gaussian approximation of the sequence of successive sums of independent random vectors with nite exponential moments is obtained.

On the rate of convergence in the weak invariance principle for dependent random variables with applications to Markov chains

“…The proof relies on a progressive blocking technique (see Bernstein [2]) coupled with a triadic Cantor like scheme and on the Komlós, Major and Tusnády approximation type results for independent r.v. 's (see [22], [10], [42]), which is in contrast to approaches usually based on martingale methods.…”

“…The rates of convergence in the weak invariance principle, for independent r.v. 's have been obtained in Prokhorov [31], Borovkov [3], Komlós, Major and Tusnády [22], Einmahl [10], Sakhanenko [34], [35], Zaitsev [42] among others. For δ ≤ 1 2 , in the case of martingale-differences, the rates are essentially the same as in the independent case (see, for instance Hall and Heyde [19], Kubilius [23], Grama [11]).…”

A New Way to Obtain Estimates in the Invariance Principle

“…If the random variables {X k } and {Y k } have moments only of the order r ≤ 3 (see (3.5)), then the estimates, obtained in Corollaries 3.1-3.3, are unimprovable in some sense, because they are unimprovable in the one-dimensional case (see, for example, Sakhanenko (1974)). But if these random variables have moments of the order r > 3, then the famous but complicated method of Komlós-Major-Tusnády and its more complicated generalizations (see Zaitsev (1998), for example) allow one to obtain incomparably better estimates for sequences of independent random variables. Moreover, the less complicated method of Einmahl (1987) also gives better estimates in case r > 3.…”

A new strong invariance principle for sums of independent random vectors

“…Moreover, it was shown in this article that under an extra smoothness assumption on the distribution of X strong approximations with even better rates, especially with rate O(log n) are possible in higher dimensions as well. Zaitsev [16] finally showed that such constructions are also possible for random vectors which do not satisfy the extra smoothness condition so that we now know that all the results of [10] have versions in higher dimensions.…”

“…In this case the problem becomes more delicate since one has to use truncation arguments which lead to random vectors with possibly very irregular covariance matrices. Most of the existing strong approximation techniques for sums of independent random vectors require some conditions on the ratio of the largest and smallest eigenvalues of the covariance matrices (see, for instance, [4,16]) and, consequently, they cannot be applied in this case. Here a new strong approximation method which is due to Sakhanenko [14] will come in handy.…”