Stochastic order and convergence notation for finite-dimensional random vectors

For each positive integer $n$, let $({\mathrm{\Omega }}_n,{\mathcal{A}}_n,P_n)$ be a probability space and let $X_n:{\mathrm{\Omega }}_n\to {\mathbb{R}}^d$ be measurable into the Borel sigma-algebra generated by the Euclidean open sets, where $d\ge 1$ is fixed and ${\mathbb{R}}^d$ has the Euclidean norm $\mathrm{\parallel }\cdot {\mathrm{\parallel }}_2$ (Euclidean spheres and closed balls as subspaces of ${\mathbb{R}}^n$, Random elements and real random variables, Probability measures and probability spaces). The spaces and measures may depend on $n$; no common probability space is required.

Write $X_n=o_P(1)$ if for every $\epsilon >0$, $P_n(\mathrm{\parallel }{X}_n{\mathrm{\parallel }}_2>\epsilon )⟶0.$ Write $X_n=O_P(1)$, or say that $(X_n)$ is bounded in probability, if for every $\eta >0$ there is $M>0$ such that ${\mathrm{lim\hspace{0.17em}sup}}_{n\to \mathrm{\infty }}{P}_n(\mathrm{\parallel }{X}_n{\mathrm{\parallel }}_2>M)\le \eta \mathrm{.}$ For real-valued variables on one common probability space, $o_P(1)$ is convergence in probability to zero as defined in Convergence in probability.

The norms and tail probabilities are part of the definition. Independence is not implicit in either notation.