The Lévy-Baxter theorem for Gaussian random fields: a sufficient condition

Abstract:

For a function $F$ of some class and for a Gaussian random field $\{ X({\text {t}});\;{\text {t}}\;\epsilon {[0,\;1]^N}\}$, $F$-variation ${V_{F,n}}(X)$ is defined as ${V_{F,n}}(X) = {\sum _{\text {i}}}F({X_{{\text {i}},n}})$, where ${X_{{\text {i}},n}} = \Delta _N^s \cdots \Delta _1^sX(s{\text {i}}),\;s = {2^{ - n}}$, and $\Delta _k^s$ is the difference operator in the $k\operatorname {th}$ component of ${\text {i}} = ({i_1}, \cdots ,{i_N})$, $(1 \leqslant {i_k} \leqslant {2^n};k = 1, \ldots ,N)$. Here is presented a sufficient condition for the existence with probability $1$ of the limit of the normalization of ${V_{F,n}}(X)$ as $n \to \infty$.