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Let $G$ be a centered stationary Gaussian process indexed by the integer lattice $\mathbb Z^d$. A straightforward Borel-Cantelli argument shows that $$\limsup_{m\to\infty}\frac{1}{\sqrt{\log m}}\left(\max_{\|x\|\leq m}G(x)\right)\leq\sqrt{2d\mathrm{Var}[G(0)^2]}.$$ In general, there need not be matching lower bound (e.g., if $G(x)=G(0)$ for all $x\in\mathbb Z^d$). However, in some cases, I know that we do have a matching lower bound, for example, in the case where the $G(x)$ are all independent.

That said, I'm trying to find a reference for the following result:

Question. Denote the covariance function $C(x):=\mathrm{E}[G(0)G(x)]$. Suppose that $C(x)\to0$ as $\|x\|\to\infty$. Do we then have that $$\liminf_{m\to\infty}\frac{1}{\sqrt{\log m}}\left(\max_{\|x\|\leq m}G(x)\right)\geq\sqrt{2d\mathrm{Var}[G(0)^2]}?$$ If so, does anyone know of a reference for such a result?

I strongly suspect the result is true, in part because I do have a reference for the same statement when $G$ is a continuous stationary Gaussian process on $\mathbb R^d$.

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For $d=1$, see the paper "Maxima of stationary Gaussian processes", by Pickands (ZW 1967), Theorem 3.4; I think (but have not checked) that a similar method works for $d>1$. Maybe the book of Adler and Taylor has further references.

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  • $\begingroup$ Thanks for that! This is exactly what I was looking for in 1d. I'll look into the higher dimensional case. $\endgroup$ – user78370 Jun 19 '20 at 11:02

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