# zbMATH — the first resource for mathematics

Extensions of results of Komlós, Major, and Tusnády to the multivariate case. (English) Zbl 0676.60038
The purpose of this excellent paper is to establish extensions of the well-known Komlós-Major-Tusnády 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.
Reviewer: T.Inglot

##### MSC:
 60F15 Strong limit theorems 60F17 Functional limit theorems; invariance principles 60F10 Large deviations
Full Text:
##### References:
 [1] Berger, E, Fast sichere approximation von partialsummen unabhängiger und stationärer ergodischer folgen von zufallsvektoren, () [2] Berkes, I; Philipp, W, Approximation theorems for independent and weakly dependent random vectors, Ann. probab., 7, 29-54, (1979) · Zbl 0392.60024 [3] Bhattacharya, R.N; Rao, R.R, () [4] Borovkov, A.A; Sakhanenko, A.I, On the rate of convergence in invariance principle, (), 59-66 · Zbl 0454.60033 [5] Csörgő, M; Révész, P, () [6] Einmahl, U, (), Techn. Rep. Ser. Lab. Res. Statist. Probab. No. 88 [7] Einmahl, U, Strong invariance principles for partial sums of independent random vectors, Ann. probab., 15, 1419-1440, (1987) · Zbl 0637.60041 [8] Einmahl, U, A useful estimate in the multidimensional invariance principle, Probab. theory relat. fields, 76, 81-101, (1987) · Zbl 0608.60029 [9] Gänssler, P; Stute, W, () [10] Komlós, J; Major, P; Tusnády, G, An approximation of partial sums of independent R.V.’s and the sample D.F., I, Z. wahrsch. verw. gebiete, 32, 111-131, (1975) · Zbl 0308.60029 [11] Komlós, J; Major, P; Tusnády, G, An approximation of partial sums of independent R.V.’s and the sample D.F., II, Z. wahrsch. verw. gebiete, 34, 33-58, (1976) · Zbl 0307.60045 [12] Philipp, W, Almost sure invariance principles for sums of B-valued random variables, (), 171-193 [13] Rosenblatt, M, Remarks on a multivariate transformation, Ann. math. statist., 23, 470-472, (1952) · Zbl 0047.13104 [14] Sakhanenko, A.I, Rate of convergence in the invariance principle for variables with exponential moments that are not identically distributed, (), 4-49, [Russian] [15] Strassen, V, An invariance principle for the law of the iterated logarithm, Z. wahrsch. verw. gebiete, 3, 211-226, (1964) · Zbl 0132.12903
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.