Expectation of hitting time for simple symmetric random walk

Question

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 $\mathbb E(T)$?

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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?

Answer 1

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