 Uniform pointwise probability bounds do not bound a random supremum

A bound on ${\sup }_x\mathbb{P}(|Z_n(x)|>t)$ does not give the same bound on $\mathbb{P}({\sup }_x|Z_n(x)|>t)$. Even independent standard normal random variables each have uniformly bounded tails while their Gaussian maximum over $N$ indices grows at scale $\sqrt{\log N}$. This distinction matters in nonparametric statistics when a shrinking bandwidth permits many spatial windows.