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Principe d’invariance faible pour la fonction de repartition empirique dans un cadre multidimensionnel et melangeant. (Weak invariance principle for multivariate and mixing empirical distributions). (French) Zbl 0651.60042

Let \(\{\xi_ n,\;n\geq0\}\) be a strictly stationary \(R^ d\)-valued process. Suppose \(d=1\). The authors obtain inequalities of Marcinkiewicz- Zygmund type for even order moments of partial sums of the process \(\{\xi_ n\}\) assuming that it is either a strong-mixing or a uniformly mixing (\(\varphi\)-mixing) process. They also derive an exponential inequality of Bernstein-type when the process is geometrically \(\varphi\)- mixing, that is, the mixing coefficient \(\phi_ n\leq a\theta^ n\), \(a\geq0\), \(0\leq\theta<1\).
Assume that \(d\geq1\). Suppose \(F_ n\) denotes the empirical distribution function corresponding to \(\{\xi_1,\ldots,\xi_ n\}\) under the usual partial ordering on \(R^ d\) and \(\{\xi_ n\}\) is geometrically \(\varphi\)-mixing. Let \(F\) be the distribution function of \(\xi_1.\)
The authors prove that there exists a sequence of zero-mean stationary Gaussian processes \(Y_ n\) with appropriate covariance function such that \[ P\left\{ \sup_{R^ d}\left| \sqrt n (F_ n - F) - Y_ n \right| \geq cn^{-a}\log n \right\} \leq cn^{-a}\log n \] where \(a=1/(3(5d+4))\) and \(c\) is a constant depending on \(d\) and \(\theta\). Similar results were obtained for strong-mixing and \(\varphi\)-mixing processes.
Reviewer: B.L.S.Prakasa Rao

MSC:

60F17 Functional limit theorems; invariance principles
60G15 Gaussian processes
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