Doukhan, Paul; Portal, Frederic 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 Probab. Math. Stat. 8, 117-132 (1987). 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 Cited in 18 Documents MSC: 60F17 Functional limit theorems; invariance principles 60G15 Gaussian processes Keywords:inequalities of Marcinkiewicz-Zygmund type; strong-mixing; uniformly mixing; exponential inequality; phi-mixing processes PDFBibTeX XMLCite \textit{P. Doukhan} and \textit{F. Portal}, Probab. Math. Stat. 8, 117--132 (1987; Zbl 0651.60042)