# Independence of the increments of Gaussian random fields

```@article{Inoue1982IndependenceOT,
title={Independence of the increments of Gaussian random fields},
author={Kazuyuki Inoue and Akio Noda},
journal={Nagoya Mathematical Journal},
year={1982},
volume={85},
pages={251 - 268}
}```
• Published 1 March 1982
• Mathematics
• Nagoya Mathematical Journal
Let be a mean zero Gaussian random field (n ⋜ 2). We call X Euclidean if the probability law of the increments X(A) − X(B) is invariant under the Euclidean motions. For such an X, the variance of X(A) − X(B) can be expressed in the form r(|A − B|) with a function r(t) on [0, ∞) and the Euclidean distance |A − B|.
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## References

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• Akio Noda
• Mathematics
Nagoya Mathematical Journal
• 1975
We shall consider the class of Gaussian random fields Xα = {X(A); A ∈ R n } such that E{X(A) − X(B)} = 0 and E{(X(A) − X(B))2} = |A − B| α (0 < α < 2), where |A − B| denotes the Euclidean distance
A spectral theory for certain types of random fields and random generalized fields (multidimensional random distributions) in the Euclidean n-space \$R_n \$ similar to the well-known spectral theories for stationary random processes is established.
THE theory of Fourier integrals arises out of the elegant pair of reciprocal formulæThe Laplace TransformBy David Vernon Widder. (Princeton Mathematical Series.) Pp. x + 406. (Princeton: Princeton