What's the definition of $O_{p}(\cdot)$ in the vector case?

Question

For a scalar random sequence $X_n$, we write $X_n=O_p(a_n)$ if for every $\epsilon$ there exists $M_{\epsilon}$ such that $\limsup_{n}\Pr(|X_n/a_n|>M_{\epsilon})<\epsilon$.

What's the extension of this definition to the vector case? For example, when we write random vector $Z_n=O_{p}(a_n)$, do we mean that for every $\epsilon>0$ there exists $M_{\epsilon}$ such that $\limsup_{n}\Pr(||Z_n/a_n||>M_{\epsilon})<\epsilon$, where $||\cdot||$ is the Euclidean norm?

Answer 1

With vectors of fixed finite dimension, say $m$, you can do this in various equivalent ways. Write $X_{ni}$ for the components of $X_n$.

1. $X_n=O_p(1)$ when the components $X_{ni}$ of $X_n$ are $O_p(1)$ as scalar sequences.
2. $X_n=O_p(1)$ when the Euclidean norm $\|X_n\|_2=\sqrt{\sum_{i=1}^m X_{ni}^2}$ is $O_p(1)$ (note: this is the random quantity $\|X_n\|_2$, not the possibly non-existent $E[\|X_n\|_2])$
3. Like 2, but for any $p$-norm $\|X_n\|_p=\sqrt[p]{\sum_{i=1}^m |X_{ni}|^p}$, for $0<p\leq \infty$

These are all equivalent to each other, and they are equivalent to tightness of the sequence, which is a topological condition: for any $\epsilon$ there exists a compact set $K_\epsilon$ such that $P(X_n\in K_\epsilon)>1-\epsilon$ for all $n$.

These are all different if the dimension is not finite or is allowed to grow with $n$; in that context you would usually be explicit and say $\|X_n\|_2=O_p(1)$ or $X_{ni}=O_p(1)$ componentwise. In particular, definition 1 is a very weak condition for infinite-dimensional vectors.