Let $X_n$ be a sequence of random variables. Show that there exist a sequnce $c_n$ of positive real numbers such that $\frac{X_n}{C_n}$ that converge to 0 almost surely.

So far, I consider the sequence $c_n$ to be some function of the expected values of $X_n$.

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    $\begingroup$ Your idea is not so great, since you cannot assume that the $X_n$ even have expected values (i.e. that they are integrable). $\endgroup$ Nov 3, 2019 at 21:27

1 Answer 1


Hint: for any numbers $\epsilon_n, \delta_n > 0$, show that you can find $C_n$ sufficiently large that $$P\left(\left|\frac{X_n}{C_n}\right| \ge \delta_n\right) < \epsilon_n.$$ Now decide how to choose $\epsilon_n, \delta_n$, thinking about the Borel-Cantelli lemma.


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