In this paper we consider processes Xₜ with values in , p ≥ 1 on subsets T of a unit cube in ℝⁿ satisfying a natural condition of boundedness of increments, i.e. a process has bounded increments if for some non-decreasing f: ℝ₊ → ℝ₊ ||Xₜ-Xₛ||ₚ ≤ f(||t-s||), s,t ∈ T. We give a sufficient criterion for a.s. continuity of all processes with bounded increments on subsets of a given set T. This criterion turns out to be necessary for a wide class of functions f. We use a geometrical Paszkiewicz-type characteristic of the set T. Our result generalizes in some way the classical theorem by Kolmogorov.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-bc90-0-7,
author = {Jakub Olejnik},
title = {On Paszkiewicz-type criterion for a.e. continuity of processes in $L^{p}$-spaces},
journal = {Banach Center Publications},
volume = {89},
year = {2010},
pages = {103-110},
zbl = {1215.60028},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-bc90-0-7}
}
Jakub Olejnik. On Paszkiewicz-type criterion for a.e. continuity of processes in $L^{p}$-spaces. Banach Center Publications, Tome 89 (2010) pp. 103-110. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-bc90-0-7/