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 $\mathbb R^d$.



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


