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Central Limit Theorem in Hölder Spaces

Račkauskas A., Suquet C.

Probability and Mathematical Statistics

1999

Vol. 19, Fasc. 1

133--152

Stochastic processes are considered within the framework of Hölder spaces Hα0 as paths spaces. Using Ciesielski's isomorphisms between Hα0 and sequences spaces via the Faber Schauder triangular functions allows us to express our basic assumptions in terms of second differences of the processes, giving more flexibility. We obtain general conditions for the existence of a version with paths in Hα0 and the tightness of sequences of random elements in these spaces. Central limit theorems in Hα0 are established and convergence rates are given with respect to Prohorov and bounded Lipschitz metrics. As an application, we study the weak Hölder convergence of the characteristic empirical process.