Summary
We give weak invariance principles for the empirical measure of a stationary strongly mixing sequence \((\xi _k )_{k \geqq 0,} X_n (f) = \frac{1}{{\sqrt n }}\sum\limits_{k = 1}^n {(f(\xi _k ) - Ef(\xi _k ))} \). For the case where f∈B s ,the unit ball of the Sobolev space H s (X) of a riemannian compact manifold, and f is a Lipα function (\((\frac{1}{2} < \alpha \leqq {\text{1}})\)) we obtain logarithmic rates of convergence ɛ n such that, for a stationary sequence of gaussian processes Y n ,\(\mathbb{P}(\mathop {\sup }\limits_f |X_n (f) - Y_n (f)| \geqq \varepsilon _n {\text{)}} \leqq \varepsilon _n \). We also prove, for the case of kernel estimates \(\hat g_n \), the existence of a gaussian non stationary sequence of random processes (Y n (x)) x∈K indexed be a compact subset K of \(\mathbb{R}^{^d } \) and constants a, b, c>0 such that
finally we give estimates of the kind:
Here h n is the window of the kernel estimate \(\hat g_n \).
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Doukhan, P., Leon, J.R. & Portal, F. Principes d'invariance faible pour la mesure empirique d'une suite de variables aléatoires mélangeante. Probab. Th. Rel. Fields 76, 51–70 (1987). https://doi.org/10.1007/BF00390275
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DOI: https://doi.org/10.1007/BF00390275