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Convergence en loi des H-variations d’un processus gaussien stationnaire sur \({\mathbb{R}}\). (Convergence in law of H-variations of a stationary Gaussian process). (French) Zbl 0691.60017

Let \(\{X,k\geq 1\}\) be a sequence of independent random variables and put \[ S_ n=\sum^{n}_{k=1}X_ k,\quad Z_ n=B_ n^{-1}S_ n-a_ n, \] where \(B_ n>0\) and \(a_ n\) are constants. We assume that \(X_ k\), \(k\geq 1\), has a density function \(f_{X_ k}\). Suppose that \[ \sup_{x}\| F_{Z_ n}(x)-G(x)| \to 0,\quad n\to \infty, \] where G is a distribution function of class L. The main result gives conditions under which \(\sup | f_{Z_ n}(x)-g(x)| \to 0,\) \(n\to \infty,\) where \(g(x)=G'(x).\)
Reviewer: D.Szynal

MSC:

60F05 Central limit and other weak theorems
60G15 Gaussian processes
62M99 Inference from stochastic processes
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