TITLE: Expectation of hitting time for simple symmetric random walk
QUESTION [6 upvotes]: Assume there is a simple symmetric random walk
$$S_n=X_1+...+X_n,\quad S_0=0$$
where $\mathbb P(X_i=\pm 1)=\frac{1}{2}$.
Define $T=\inf\{n:S_n=1\}$. How to compute $\mathbb E(T)$?

My idea: if $\mathbb E(T)<\infty$ then
$$\mathbb E(S_T)=\mathbb E(T)\mathbb E(X_i)$$
where $\mathbb E(S_T)=1$, $\mathbb E(X_i)=0$  so there is a contradiction.
Therefore, $\mathbb E(T)=\infty$. 
Is there something wrong?

REPLY [2 votes]: Your proof is correct: the expected value of this hitting time is infinite. 
To add, $S_n$ is a Markov Chain (symmetric random walk) on the integers. It is null recurrent. The proof you just gave verifies it cannot be positively recurrent. -- Lost1