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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



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

Author: Takayuki Kawada
Journal: Proc. Amer. Math. Soc. 53 (1975), 463-469
MSC: Primary 60G15
MathSciNet review: 0383512
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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 $.

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Keywords: Gaussian random field, mixed-increment, $ F$-variation, strong limit of $ F$-variation
Article copyright: © Copyright 1975 American Mathematical Society

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