On smoothness conditions for trajectories of random functions

Let $\xi(x)$ be a random function of $x\in R^k$ and
$$ \omega_p^r(\delta,\xi)=\sup_{\substack{x,h\in R^k\\|h|\le\delta}} \mathbf E^{1/p}\biggl|\sum_{l=0}^r(-1)^lC_r^l\xi(x+lh)\biggr|^p. $$
Properties of $\omega_p^r(\delta,\xi)$ as a function of $\delta$ are investigated; a number of inequalities are obtained.