# Complete convergence is equivalent to convergence a.s. under independence

$X_1,X_2,\ldots$ is a sequence of random variables that are complete convergent to $X$ if $$\sum_{n=1}^{\infty} P(\mid X_n-X\mid >\epsilon)<\infty \space\forall\epsilon > 0$$. Show if $X_n$ are independent, then complete convergence is equivalent to convergence a.s.

I showed that complete convergence implies convergence a.s. using the Borel-Cantelli lemma, but I'm not sure how to use show the converse using independence. This is what I have so far:

$$X_n\rightarrow_{a.s.} X \implies X_n\rightarrow_{p} 0$$ WLOG, $X=0$.

$$\forall\epsilon >0, P(\mid X_n\mid > \epsilon)\rightarrow 0$$ $$P(\limsup_{n\rightarrow\infty} \mid X_n\mid > \epsilon)=0$$

Since they are independent, then $\{\mid X\mid > \epsilon\}$ are independent, and can I use Borel-Cantelli (ii) to say $\sum P(\mid X_n\mid >\epsilon) < \infty$?

• en.m.wikipedia.org/wiki/Borel–Cantelli_lemma has a converse result using independence, but that you might have to prove it anyway.
– Evan
Commented Sep 28, 2013 at 18:54
• Your proof looks fine. Commented Sep 28, 2013 at 19:23
• a.s. = almost surely (probability theory) Commented Nov 15, 2022 at 21:57

Suppose that you do have the a.s. convergence but not the complete convergence. Therefore you have: $$\sum_{n=1}^{\infty} P(\mid X_n-X\mid >\epsilon)=\infty \space\exists\epsilon > 0$$ For such $\epsilon$, you can use second Borel-Cantelli to prove that: $$P(\limsup_{n\rightarrow\infty} \mid X_n-X\mid > \epsilon)=1$$ But from a.s. convergence you have $\displaystyle P(\lim_{n\rightarrow\infty} X_n=X)=1$ which means that for all $\epsilon>0$, $\displaystyle P(\limsup_{n\rightarrow\infty} \mid X_n-X\mid > \epsilon)=0$ which is a contradiction. Therefore $X_n$ should have complete convergence.
• how does $X_n\rightarrow_{a.s.}X$ imply $\sum P(|X_n-X|>\epsilon)=\infty$? Commented Sep 28, 2013 at 22:12
I think that the conclusion to your proof could be: $$P(\limsup \ |X_n|> \epsilon) = 0 \to P(\liminf \ |X_n|< \epsilon) = 1 \to P(\cup\cap |X_n| < \epsilon)=1$$ Now: $$\lim_{n \to \infty}P(\cap_{m>n \ } |X_n| < \epsilon)=1 \to \lim_{n \to \infty}P( |X_n| < \epsilon)=1$$ using the independence, that is the intersection could become a infinity product of probability, therefore $$p^{\infty}$$ but taking the reciprocal power both to $$p$$ and to the rhs (that is $$1$$), we have the result.