Consider a centered Gaussian random field on $\mathbb{R}^n$ with continuous covariance and a.s. continuous sample paths. What is known about the growth of the sample paths at infinity of such a random field?

If the random field is assumed in addition to be stationary, it appears, based on the exponential decay of the tail probabilities, that the growth should be logarithmic (as it is in the one-dimensional case) but is that written up somewhere? (The one-dimensional case is treated in Theorem 1.4. in this article: https://www.jstor.org/stable/2239990).

Without assuming stationarity, I'm interested if the following is true: Given that the covariance has polynomial growth, is it true that sample paths have a.s. polynomial growth?

5more comments