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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?

  1. 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).

  2. 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?

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  • $\begingroup$ Do you mean stationary field? $\endgroup$
    – ofer zeitouni
    Commented Mar 10, 2022 at 10:57
  • $\begingroup$ There are results like by Colella and Lanford. See for example this paper by Rosen and Simon which mention almost sure $\sqrt{\log|x|}$ type growth projecteuclid.org/journals/duke-mathematical-journal/volume-42/… $\endgroup$
    – Abdelmalek Abdesselam
    Commented Mar 10, 2022 at 14:27
  • $\begingroup$ @oferzeitouni, I suppose to get logarithmic growth, the field should be stationary or close to stationary. But I'm more interested in this: Given that the covariance has polynomial growth, is it true that sample paths have a.s. polynomial growth? $\endgroup$
    – S.Z.
    Commented Mar 10, 2022 at 15:55
  • $\begingroup$ @AbdelmalekAbdesselam, thanks for the references, these do address sample path growth but the context seems to be rather different. I'm more interested in multi-dimensional version of a result like Theorem 1.4 in this article: jstor.org/stable/2239990 $\endgroup$
    – S.Z.
    Commented Mar 10, 2022 at 16:23
  • $\begingroup$ Maybe the context is not familiar to you but it is not different. The papers I mentioned concern random distributions in $S'(\mathbb{R}^n)$ but if you mollify them by convolution then you get one of the stationary multidmensional Gaussian processes you are considering. Namely your $X(t), t\in\mathbb{R}^n$ is $q(f(\cdot-t))$ in the notations of Rosen and Simon. Also note their result is sharper than the upper bound in Thm 1.4 of Marcus. $\endgroup$
    – Abdelmalek Abdesselam
    Commented Mar 10, 2022 at 18:17

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I will consider stationary Gaussian processes $X_v$ indexed by $v\in Z^d$, not continuous time (the argument for continuous time requires a bit of extra work, and some assumptions on the short-time behavior of the covariance; smoothness of the covariance at $0$ should be enough). In general, good references on such questions are the lecture notes of R. Adler, or his book with Taylor.

By the Royen Gaussian correlation inequality $$ P(\sup_{v\in Z^d: |v|_\infty<T} |X_v|<R)\geq \prod_{v\in Z^d:|v|_\infty<T} P(|X_v|<R)= \big(P(|X_0|<R)\big)^{(2T)^d}$$ from which you get a logarithmic behavior (that is, if $R> C_d \sqrt{\log T}$ then the probability above goes to $1$, with explicit $C_d$). The same bound can also be obtained from a union bound. A complimentary lower bound requires more work but with fast enough (any polynomial) decay of correlation, also holds.

A similar argument works for non-stationary fields, but of course the answer depends on the rate of growth of the covariance (and the move from discrete to continuous requires a bit more assumptions). In general, up to extra log factors, the growth of the variance (not covariance, as soon as the latter decays fast enough, e.g. polynomially) determine the behavior.

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