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 \} $
Show that $ S_{ min(T,n) } $ converges almost surely towards a random variable X but that it does not converge in $ \mathbb{L}^1 $. What is the distribution of X?
My first guess is that this convergence occurs iff $ \: \: \mathbb{P} ( T < \infty ) = 1 $ but how do we compute this probability? We could try to find an upper bound for $ \: \: \mathbb{P} ( T = \infty ) $ ? Borel-Cantelli doesn't seem to work here either.
If that works then we could use Doob's convergence theorem.
Thanks for any insight on this.
Best Answer
Hints: