rate of convergence of the a.s. zero series

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 $$n^{s}X_{n} =O_{p}(n^{s})$$?