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Principes d’invariance faible pour la mesure empirique d’une suite de variables aléatoires mélangeante. (Weak invariance principles for the empirical measure of a mixing sequence of random variables). (French) Zbl 0596.60037

We give weak invariance principles for the empirical measure of a stationary strongly mixing sequence \((\xi_ k)_{k\geq 0}\), \[ X_ n(f)=(1/\sqrt{n})\sum^{n}_{k=1}(f(\xi_ k)-Ef(\xi_ k)). \] For the case where \(f\in B_ s\), the unit ball of the Sobolev space \(H_ s(X)\) of a Riemannian compact manifold and f is a \(Lip_{\alpha}\) function \((<\alpha \leq 1)\), we obtain logarithmic rates of convergence \(\epsilon_ n\) such that, for a stationary sequence of Gaussian processes \(Y_ n\), \[ {\mathbb{P}}(\sup | X_ n(f)-Y_ n(f)| >\epsilon_ n)\leq \epsilon_ n. \] We also prove, for the case of kernel estimates \(\hat g_ n\), the existence of a Gaussian nonstationary sequence of random processes \((Y_ n(x))_{x\in K}\) indexed be a compact subset K of \({\mathbb{R}}^ d\) and constants \(a,b,c>0\) such that \[ {\mathbb{P}}(\sup_{x\in K}| n h^ d_ n)^{1/2} (\hat g_ n(x)- g(x))-Y_ n(x)| \geq c\quad n^{-a})\leq cn^{-a}\quad if\quad h_ n=n^{-b}; \] finally we give estimates of the kind: \(E\sup_{x\in K}(\hat g_ n(x)-g(x))^ 2\leq cn^{-a}\) if \(h_ n=n^{-b}\). Here \(h_ n\) is the window of the kernel estimate \(\hat g_ n\).

MSC:

60F17 Functional limit theorems; invariance principles
60G15 Gaussian processes
62G05 Nonparametric estimation
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