Geodesic
On smoothness conditions for trajectories of random functions
Teoriâ veroâtnostej i ee primeneniâ, Tome 28 (1983) no. 2, pp. 229-250

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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. 
@article{TVP_1983_28_2_a2,
     author = {I. A. Ibragimov},
     title = {On smoothness conditions for trajectories of random functions},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {229--250},
     publisher = {mathdoc},
     volume = {28},
     number = {2},
     year = {1983},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1983_28_2_a2/}
}
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I. A. Ibragimov. On smoothness conditions for trajectories of random functions. Teoriâ veroâtnostej i ee primeneniâ, Tome 28 (1983) no. 2, pp. 229-250. http://geodesic.mathdoc.fr/item/TVP_1983_28_2_a2/