# First order asymptotics for maxima of stationary Gaussians with vanishing covariance

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 $$\mathbb R^d$$.

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.