I posted this on mathstackexchange several days ago, but it may be more appropriate to ask this here.

Let $r_n$ be some real sequence converging to zero. Let $X_n$ be a sequence of random variables. I know that $X_n=O_p(r_n)$, which means that $\forall\varepsilon>0,\exists M$ such that $$\Pr(|X_n|>r_nM)<\varepsilon,\quad \forall n$$

Does this imply that $\forall \varepsilon>0$, there exists some finite constant $C$ such that $$\Pr(|X_n|>\varepsilon)\leq Cr_n?$$

It is clear that since $r_n\to 0$, $X_n\to_p0$, so that $\forall\varepsilon>0$ $$\lim_{n\to\infty}\Pr(|X_n|>\varepsilon)=0,$$ but I don't see how to obtain what I need.


1 Answer 1


Consider $X_n$ such that $\mathbb P\left(X_n=n\right)=1/n$ and $\mathbb P\left(X_n=0\right)=1-1/n$, and $r_n:=1/n^2$. Since the sequence $X_n/r_n$ converges to $0$ in probability, we have in particular $X_n=O_p(r_n)$. However, for each positive $\varepsilon$, the quantity $\mathbb P\left(X_n\gt\varepsilon\right)$ is equal to $1/n$ if $n\gt \varepsilon$.


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