Stochastic order ($O_{\mathbb{P}}(a_n)$)

The notation $Z_n=O_{\mathbb{P}}(a_n)$ means that $Z_n/a_n$ is bounded in probability asymptotically: for every $\epsilon >0$ some $C$ gives ${\mathrm{limsup}}_n\mathbb{P}(|Z_n|>C{a}_n)\le \epsilon$. A tail bound $\mathbb{P}(|Z_n|>z{a}_n)\le C_0/z^2$ is sufficient. In statements involving bandwidths, specify whether the constants must be uniform over bandwidth sequences.

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.