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In order to solve a puzzle in correspondence with a friend, I am using a simple random walk hitting time calculation based upon the reflection principle as it is expressed in this University of Chicago document, here. However the author skips over a step and I haven't been able to fill in the gap.

Let $S_n$ be a random walk starting at zero and $\tau(m)$ be the first time it reaches $m\geq1$. Then define a new path, for $n\geq0$: \begin{align} S_n^*= \begin{cases} S_n, & \text{if }n\leq \tau(m) \\ 2m-S_n, & \text{if }n> \tau(m) \end{cases} \end{align} It is easy to see that $S_n^*$ is also a random walk. In the event $\tau(m)< n$, the two walks are on opposite sides unless they are both on $m$. We are interested in the event $S_n=m+k$ with $k\geq0$, which depends upon $\tau(m)< n$ to be non zero in probability. Thus: \begin{align} P(S_n=m+k) \begin{cases} 0, & \text{if }n\leq \tau(m) \\ P(S_n=m+k, \tau(m)< n), & \text{if }n> \tau(m) \end{cases} \end{align} This is equivalent in probability to $P(S_n^*=m+k, \tau(m)< n)$, which is equivalent in position to $P(S_n=m-k, \tau(m)< n)$. Therefore we can marginalize out $P(S_n=m+k)$ to compute the marginal probability $P(\tau(m)< n)$ as follows. \begin{align} P(\tau(m)< n)=\sum_{k=-\infty}^{\infty}P(S_n=m+k, \tau(m)< n)=P(S_n=m)+2P(S_n >m) \end{align} It is the very last equality that I don't see an obvious approach to. \begin{align} \sum_{k=-\infty}^{\infty}P(S_n=m+k, \tau(m)< n)=P(S_n=m)+2P(S_n >m) \end{align}

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Note that $$P(S_n\gt m,\tau(m)\lt n)=P(S_n\lt m,\tau(m)\lt n),$$ which explains all the sum except the $k=0$ term, hence the identity holds if and only if $$P(S_n=m,\tau(m)\lt n)=P(S_n=m), $$ which is wrong in general. My guess is that the question is to show \begin{align} P(\tau(m)\leqslant n)=\sum_{k=-\infty}^{\infty}P(S_n=m+k, \tau(m)\leqslant n)=P(S_n=m)+2P(S_n \gt m), \end{align} which holds thanks to the first identity above and $$P(S_n=m,\tau(m)\leqslant n)=P(S_n=m), $$ since $[S_n=m]\subseteq[\tau(m)\leqslant n]$.

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You are correct. $P(\tau(m)<n)$ is $P(\tau(m)\leqslant n)$ in the original document. I had assumed it was an error but I can see now that it wasn't. –  Meadowlark Bradsher Nov 11 '12 at 23:52

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