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 \(\delta\to 0\). 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 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.


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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