A tightness criterion in Besov-Orlicz spaces and applications to the problem of occupation times
The authors give a brief review of fractional derivatives (in one dimension) and Besov-Orlicz spaces, in particular the Haar-Schauder wavelet representation due to \textit{Z. Ciesielski, G. Kerkyacharian} and \textit{B. Roynette} [Stud. Math. 107, No. 2, 171--204 (1993; Zbl 0809.60004)]. Then they derive a tightness result for(sequences of) processes in Besov-Orlicz spaces. This leads to a further proof of Kolmogorov-Chentsov-type path regularity theorems. This result is subsequently applied to Besov-Orlicz regularity of Gaussian processes, of local times of rotationally symmetric \(\alpha\)-stable Lévy processes and their fractional (in time) derivatives. Finally, limit theorems for expressions of the form \(\lambda^{(\alpha-1-\gamma)/\alpha}\int_0^{\lambda t} f(X_s)\,ds\), \((\lambda\to\infty)\) where \(f\) is a fractional derivative of some Hölder function, in Besov norm are established.
scientific article; zbMATH DE number 5989456
scientific article