Continuing this post,
We consider a simple symmetric random walk $ (S_n)_{n \in \mathbb{N}} $ on $ \mathbb{Z} $ which starts at 1 : $ S_0 = 1 $ and there exists an iid sequence $ (X_n)_{n \geq 1} $ such that $ \mathbb{P}(X_1 = -1) = \mathbb{P}(X_1 = 1) = 1/2 $ and $ \forall n \in \mathbb{N} $ $ S_{n+1} = S_n + X_{n+1} $ And we look at the stopping time $ T = inf \{ n \geq 0, S_n = 0 \} $
I am curious how one computes $E(T)$, or even show it is finite/infinite? (I seem to remember I have read that $E(T)=\infty$.)
My idea:
$E(\tau) = \int_{\tau=n} \tau(\omega) \, dP = \sum n P(\tau =n) =\sum P(\tau \ge n)$. By Borell Cantelli's, if the latter is infinite,then $P( \{\tau \ge n\} \, i.o. )=1$, contradicting $P(\tau < \infty)=1$. But this holds only if the events are independent, which are clearly not...