I have the following problem.
Let $(p_n)$ be a sequence of real numbers in $[0,1]$ such that $p_n\rightarrow 0$. We take $(X_n)$ be a sequence of random variables s.t. $$\Bbb{P}(X_n=0)=1-p_n,~\Bbb{P}(X_n=1)=p_n$$Show that if $\sum_n p_n <\infty$ the sequence converges a.s. to $0$.If we also suppose that the $(X_n)$ are independent, show that this condition is also necessairy.
I somehow don't see how to show that this is necessairy, so what do I exaclty need to prove?
Thanks for your help
This is a consequence of the Borel-Cantelli Theorem. If $\sum_n P(X_n = 1) < \infty$ then $P(X_n = 1 \mbox{ infinitely often} ) = 0$ by the first Borel-Cantelli Theorem, hence $X_n \to 0$ almost surely. When the events $\{X_n =1\}$ are independent, there is a converse to Borel Cantelli Theorem (the ``second" Borel Cantelli Theorem) which adapted to this context gives that if $\sum_n P(X_n = 1) = \infty$, then $P(X_n = 1 \mbox{ infinitely often})=1$, so $X_n \not\to 0$ a.s.. Hence the condition $\sum_n P(X_n = 1)< \infty$ is necessary in order for $X_n \to 0$ a.s. when the $X_n's$ are independent.