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Given two sequence $(X_n),(Y_n)$ of random variables with values in $\mathbb{R}^d$, consider the following definitions:

  1. Write $X_n=O_p(Y_n)$ if, for all $\epsilon>0$, there exists $M>0$ and $N\geq1$ such that $P\Big[\|X_n\|>M\|Y_n\|\Big]<\epsilon$ whenever $n\geq N$.

  2. Write $X_n=o_p(Y_n)$ if, for all $\epsilon>0$, $P\Big[\|X_n\|>\epsilon\|Y_n\|\Big]\to 0 \text{ as } n\to \infty.$

In the book Asymptotic Statistics, A.W. van der Vaart defines $X_n=O_p(Y_n)$ if $X_n=Z_nY_n$ with $Z_n=O_p(1)$, and $X_n=o_p(Y_n)$ if $X_n=Z_nY_n$ with $Z_n=o_p(1)$.

But these definitions are not equivalent to the definitions $1,2$ above. For if we take $Y_n=0$ for all $n$, then $1,2$ are equivalent to $X_n=0$ with probability approaching $1$, while A.W. van der Vaart's definitions are equivalent to $X_n=0$ for all $n$.

Which of these definitions are preferable? It seems to me that $1,2$ are better since they reduce to the usual meanings of $o,O$ in the nonstochastic case (see here for example).

Am I missing something? Thanks a lot for any help.

EDIT. Van der Vaart's definition of $X_n=O_p(1)$ is, for all $\epsilon>0$, there exists $M>0$ such that $\sup_{n\geq 1} P[\|X_n\|>M] <\epsilon$. Using the fact that any real random variable is tight (see here), we see that this definition is equivalent to definition $1$ with $Y_n=1$ for all $n$. Van der Vaart's definition of $X_n=o_p(1)$ is $X_n\overset{p}\to 0$ as $n\to\infty$.

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I think the issue is maybe that Van der Vaart does not exactly spell out what is meant by $O_p(1)$ and $o_p(1)$ he just notes in the paragraph above that the first means bounded in probability and the second means convergence to $0$ in probability.

However, I think the effective definition that he uses for $O_p(1)$ is that $\Pr[||X_n||\leq M_\epsilon]>1-\epsilon$ for sufficiently large $n$ which should be equivalent to your definition. And similarly $o_p(1)$ should be taken to mean for all $\epsilon>0$ that $\Pr[||X_n||>\epsilon]\to 0$ as $n\to\infty$. I think of course these make sense because we don't want this to be true for all $n$ but for sufficiently large $n$.

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  • $\begingroup$ Van der Vaart's definitions of $O_p(1)$ and $o_p(1)$ are indeed the usual ones (see my edit). My point is that his definitions of $O_p(Y_n)$ and $o_p(Y_n)$ are not equivalent to definitions $1,2$, and these definitions seem very natural to me. $\endgroup$
    – Alphie
    Commented Jun 4, 2021 at 13:43

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