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Invariance principles in Hölder spaces. (English) Zbl 0965.60011
The weak convergence of random elements in Hölder space $H_\alpha[0,1]$ is studied, especially in a separable subspace $H^0_\alpha$ of functions $f$, for which the Hölderian modulus of continuity $\omega_\alpha (f, \delta)\to 0$, as $\delta\to 0$. {\it J. Lamperti} [Trans. Am. Math. Soc. 104, 430-435 (1962; Zbl 0113.33502)] established a Hölderian version of i.i.d. Donsker-Prokhorov’s invariance principle. In the present paper it is extended to the cases of strong mixing and of associated random variables, see the moment inequalities for the latter of {\it T. Birkel} [Ann. Probab. 16, No. 3, 1184-1193 (1988; Zbl 0647.60039)]. Similar convergence is proved for the convolution smoothing of partial sums processes. The presentation is very clear. Some proofs of auxiliary statements are given, which are known in the literature.

##### MSC:
 60B10 Convergence of probability measures 60F05 Central limit and other weak theorems 60F17 Functional limit theorems; invariance principles 62G30 Order statistics; empirical distribution functions
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