Probability bound for the tail of a random serie Let $X_n$ a random sequence such that $E(X_n) = 0$ and $E(X_n^2)= 1$ If $a_n$ is a positive decreasing sequence such that $\sum_{n=1}^\infty a_n < \infty$. Is the following statement abount the tail of the random serie true? 
$$S_k = \sum_{n=k+1}^\infty a_n X_n = \mathcal{O}_P(\sum_{n=k+1}^\infty a_n)$$
I know by Tchebichef inequality that all $X_n$ are bounded in probabilty with the same constant. But I am not sure if that is enough.

Let $r_k:=\sum_{n=k+1}^\infty a_n$. Then by Markov's inequality, 
$$
\mathbb P\left(\left\lvert S_k\right\rvert /r_k\gt M\right)\leqslant M^{-1}r_k^{-1}\mathbb E\left\lvert S_k\right\rvert\leqslant M^{-1}r_k^{-1}\sum_{n=k+1}^\infty a_n\mathbb E\left\lvert X_n\right\rvert
$$
and by Cauchy-Schwarz inequality, $\mathbb E\left\lvert X_n\right\rvert\leqslant 1$ for all $n$ hence $\mathbb P\left(\left\lvert S_k\right\rvert /r_k\gt M\right)\leqslant M^{-1}$.
