Uniform pointwise probability bounds do not bound a random supremum
= 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.