Invariance Principles in Hölder Spaces

Djamel Hamadouche

Abstract: We study the weak convergence of random elements in the space of Hölder functions $\H_\alpha[0,1]$. Using this space instead of $C[0,1]$ enables us to obtain functional limit theorems of a wider scope. Some examples of Hölder continous functionals of the paths are proposed to illustrate this improvement. A new tightness condition is established. We obtain an Hölderian version of Donsker-Prohorov's invariance principle about the polygonal interpolation of the partial sums process, generalizing Lamperti's i.i.d. invariance principle to the case of strong mixing or associated random variables. Similar results are proved for the convolution smoothing of partial sums process.

Keywords: Tightness; Hölder space; triangular functions; invariance principles; strong mixing; association; Brownian motion.

Classification (MSC2000): 60B10, 60F05, 60F17, 62G30.

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