Expectation of hitting time for simple symmetric random walk

$\begingroup$

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?

$\endgroup$ 4

1 Answer

$\begingroup$

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

$\endgroup$

Your Answer

Sign up or log in

Sign up using Google Sign up using Facebook Sign up using Email and Password Submit

Post as a guest

Post Your Answer Discard

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy