# Convergence of RV sequence

Assume a sequence of random variables $Z_1,Z_2,Z_3,...$ such that $$\sum_{i=1}^{\infty} P(|Z_i-Z|>ε)<\infty$$, for$ε>0$ , then $Z_i \to Z$ almost surely as $n \to \infty$.

So I understand that if I can prove that $Z_i$ converges in probability then it converges almost surely. To prove it converges in probability, I am thinking of sequence of events $A_1\subseteq A_2\subseteq A_3\subseteq\cdots$ and $A_1$ being the $|Z_{\infty}-Z|>ε$. And then prove all $P(A_i)$ has to euqal to $0$?

## 1 Answer

No, that doesn't work. Convergence in probability does not imply almost sure convergence. The theorem you want to use is the first Borel-Cantelli lemma. Let $A_i$ be any sequence of events in a probability space. Then

$$\sum_{i=1}^{\infty} P(A_i) < \infty \implies P(A_i\ \text{infinitely often}) = 0$$

Can you complete the proof with this?

Also if you want an example where convergence in probability does not imply convergence almost surely look here:

Convergence of random variables in probability but not almost surely.

• let A1 be the event that $|Z1-Z|>\epsilon?$ – james black Apr 9 '18 at 5:55
• That's a place to start, you want to eventually end up with an event $A$ such that $\forall \omega \in A$ we have $\lim_{n \to \infty} Z_n(\omega) = Z(\omega)$ and $P(A) = 1$. – bitesizebo Apr 9 '18 at 6:05
• right the only thing im wondering bout is $A_1\subseteq A_2\subseteq A_3\subseteq\cdots$ might not be true since the question doesnt stay that Zn is a converging series so what should i do about that? thanks – james black Apr 9 '18 at 6:08
• You've not defined the events $A_i$ so I don't understand what you're trying to say here. – bitesizebo Apr 9 '18 at 6:10