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The following question occurred to me: Suppose $X_n$ is a sequence of positive random variables satisfying for all $\delta>0$, $P(X_n < \delta) \to 1$. Is it true that there must exist a sequence $a_n \to 0$ so that $X_n = O_P(a_n)$? My inclination is that this is false, but I cannot come up with a counter example. Such a counter example sequence of random variables would effectively need to decrease to zero "arbitrarily slowly". Can any of you smart folks come up with a proof or counter example?

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For any $\delta>0$ there exists $N_{\delta}\ge 1$ such that $$ \mathsf{P}(X_n\ge \delta)\le \delta $$ for all $n\ge N_{\delta}$. Thus, you may construct the required sequence $\{a_n\}$ as follows. Take $a_n=1$ for $1\le n< N_{1/2}$, $a_n=1/2$ for $N_{1/2}\le n< N_{1/4}$, etc.

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