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I have given $S_0=0$ and $S_n=X_1+...+X_n$ for all $n\geq 1$ where $X_i$ are an i.i.d. sequence such that $\Bbb{P}(X_i=-1)=\Bbb{P}(X_i=1)=1/2$. Now for $u,v\in \Bbb{N}$ fixed I want to show that $\inf\{n\in \Bbb{N}:S_n\in \{-u,v\}\}<\infty$ a.s.

I thought about using Borell-Cantelli 2 to show this. But somehow this does not work. But now I am a bit lost what to do next. I know that $\inf\{n\in \Bbb{N}:S_n\in \{-u,v\}\}$ describes the smallest $n$ such that the sum is either $-u$ or $v$, i.e. it is the first time where $S_n$ reaches $-u$ or $v$.

I also tried to use the definition, i.e. $T<\infty$ a.s. iff $\Bbb{P}(\{T<\infty\}=1$.

Could maybe someone give me a hint how to show this?

user1294729
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  • Have you checked the optional stopping theorem? Check this https://www.cl.cam.ac.uk/teaching/1819/Probablty/materials/Lecture8.pdf – KBS Oct 15 '22 at 13:43
  • @KBS But our version only tells us that if $X_n$ is a martingale ant $T$ a stopping time then $X_{T\wedge n}$ is again a martingale. But this does not tell me much about if $T$ is a.s. finite does it? – user1294729 Oct 15 '22 at 13:46
  • You may look at Markov inequalities in order to have a relationship between expectation and probabilities. – KBS Oct 15 '22 at 13:55
  • @KBS Sorry but why do I need this relation? Are you still speaking about optional stopping thm? – user1294729 Oct 15 '22 at 13:56
  • @KBS or do you think I should show $\Bbb{P}({T=\infty})=0$? – user1294729 Oct 15 '22 at 14:00
  • @KBS I have found something here but don't get the comment to the question : https://math.stackexchange.com/questions/174134/stopping-time-and-random-walk-proof-that-stopping-time-of-reaching-a-certain-va – user1294729 Oct 15 '22 at 14:13

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Let $\tau=\inf\{n:S_n \in\{-u,v\}\}$. Note $S_n^2-n$ is a martingale. By optional stopping on the bounded stopping time $n \wedge \tau=\min(n,\tau)$ we get $$0=E[S_{n \wedge \tau}^2-(n\wedge \tau)]\implies E[(n\wedge \tau)]=E[S_{n\wedge \tau}^2]\leq \max(u^2,v^2)$$ since $S_{n\wedge \tau}^2\leq \max(u^2,v^2)$ by construction of $\tau$. We conclude with monotone convergence for $n \to \infty$.

Snoop
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