5
$\begingroup$

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?

$\endgroup$
  • 1
    $\begingroup$ No. Next question? $\endgroup$ – Did Dec 30 '14 at 9:18
  • 1
    $\begingroup$ @Did oh,I just want to verify my proof,the conclusion is so strange. $\endgroup$ – Lookout Dec 30 '14 at 9:22
  • 1
    $\begingroup$ What is "so strange" here? $\endgroup$ – Did Dec 30 '14 at 9:26
  • $\begingroup$ 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. $\endgroup$ – Lost1 Jan 1 '15 at 22:49
2
$\begingroup$

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

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

Not the answer you're looking for? Browse other questions tagged or ask your own question.