Stochastic order
= Stochastic order
{title2=$O_{\mathbb P}(a_n)$}
The notation $Z_n=O_{\mathbb P}(a_n)$ means that $Z_n/a_n$ is <bounded in probability> asymptotically: for every $\varepsilon>0$ some $C$ gives $\limsup_n\mathbb P(|Z_n|>Ca_n)\leq\varepsilon$. A tail bound $\mathbb P(|Z_n|>za_n)\leq C_0/z^2$ is sufficient. In statements involving bandwidths, specify whether the constants must be uniform over bandwidth sequences.