# On the definition of the stochastic $o$ and $O$ symbols

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 $$n\to\infty$$.

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 $$n$$.
• 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. Commented Jun 4, 2021 at 13:43