Let $\{X_{n}: n\geq1\}$ be i.i.d random variables with the common distribution $\mathbb{P}(X_n=1)=p$ and $\mathbb{P}(X_n=-1)=q=1-p$ where $0<p<1$. Define $S_0:=0$ and $S_n = \sum_{j=1}^{n} X_j$ for $n\geq 1$. Then $\{S_n:n\geq0\}$ is a $(p-q)random\;walk$ on $\mathbb{Z}$ (When $p=q=\frac{1}{2}\;,\;\{S_n:n\geq0\}\;is\;a\;symmetric\;random\;walk$). Given two positive integers a and b, consider $$\tau:=inf\{n\geq1:S_n=-a\;or\;S_n=b\}$$

I want to show that $\mathbb{E}[\tau]<\infty$.

What I know and I can use is that

Given a filtered space $\left(S, \mathcal{\Sigma}, \{\mathcal{\Sigma_n: n\ge 0}\}, \mathbb{P}\right)$, and a stopping time $\tau: S \to \{0, 1,2, \ldots\}$ with respect to $\{\mathcal{\Sigma_n: n\ge 0}\}$, suppose that there exist $m\in\mathbb{N}$ and $\epsilon$ such that, for every $n\ge 0$: $$\mathbb{E}\left[\mathbb{I}_{\tau \le n+m} | \mathcal{\Sigma_n}\right] > \epsilon \quad \text{a.s.}.$$ then $\mathbb{E}[\tau]<\infty$.

And I also know that it's sufficient to show that there exist $m\in\mathbb{N}$ and $\epsilon$ such that, for every $n\ge 1$ and every $A=\{S_1=a_1,\cdots , S_n=a_n\}$ where $a_j \in \mathbb{Z}$ for $1\leq j \leq n$: $$\mathbb{P}(A \cap\{\tau\leq n+m\})\geq \epsilon \mathbb{P}(A)$$.

I'm trying to build a link between these two. Any hint is appreciated.

  • $\begingroup$ What's wrong about using martingale techniques? It's much easier... moreover, it is very likely that you are not the first to ask this question, so you might want to use the search $\endgroup$ – saz Dec 3 '18 at 21:25

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