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Let us consider the following sequence $n^{s}X_{n}$, where $X_{n}$ are random variables defined on the same probability space, and $s>0$. Assume that we know that $$ P\left(\liminf_{n\to\infty}A_n\right)=1, $$ where $A_n = \{ \omega \in \Omega: X_{n}(\omega) = 0\}$.

Does it make any sense to say something about the rate of convergence. i.e. writing $n^{s}X_{n} =O_{p}(n^{s})$?

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