an:04107844
Zbl 0676.60038
Einmahl, Uwe
Extensions of results of Koml??s, Major, and Tusn??dy to the multivariate case
EN
J. Multivariate Anal. 28, No. 1, 20-68 (1989).
00170440
1989
j
60F15 60F17 60F10
multivariate quantile transformation; strong approximations; KMT-type inequalities; large deviation theorem; conditional distribution
The purpose of this excellent paper is to establish extensions of the well-known Koml??s-Major-Tusn??dy strong approximations to the multivariate case. One of the main results can be stated as follows:
Let H be a continuous, nonnegative function on [0,\(\infty)\) such that \(H(t)/t^{3+r}\) is eventually increasing for some \(r>0\) and log H(t)/t\({}^{1/2}\) is eventually non-increasing. Suppose X is a d- dimensional random variable with mean 0, covariance matrix \(\Sigma\) and E H(\(| X|)<\infty\), where \(| \cdot |\) denotes the Euclidean norm. Then i.i.d. sequences \(\{X_ n\}\), \(\{Y_ n\}\) can be constructed in such a way that \(X_ n=^{D}X\), \(Y_ n\) is N(0,\(\Sigma)\)- distributed and
\[
T_ n=\sum^{n}_{1}X_ k-\sum^{n}_{1}Y_ k=O(H^{-1}(n))\quad a.s.
\]
If the moment generating function of X exists and satisfies a mild smoothness condition then the rate O(log n) can be achieved. Simultaneously with the above, anologues of KMT-type inequalities (e.g. exponential inequalities) for \(T_ n\) are also obtained.
The basic tool in the proofs is an extension of the quantile transformation method of KMT to the multidimensional case. To get this a large deviation theorem for conditional distribution functions is first proved.
T.Inglot