Grimmett and Stirzaker's "Probability and Random Processes" gives a nice discussion about random walk, for example, it considers
$M_n=\max\{S_i,0\le i \le n\}$ where as usual $$S_i=\sum_{k=1}^i{X_k},$$ here $S_0=0$, $\mathbb{P}(X_k=1)=p$ and $\mathbb{P}(X_k=-1)=q=1-p$.
One of the results about $M_n$ is $\mathbb{P}(M_n\ge r, S_n=b)$ or even $\mathbb{P}(M_n\ge r)$. I am just wondering how to relate to the minimum $m_n=\min\{S_i,0\le i \le n\}$ given this result? Is there any simple way to do this?
I just found sometimes considering the minimum is most convenient for some problems.
Best Answer
Let $m_n^p = \min\{S_i, 0\leq i\leq n, P(X_i=1)=p\}$, with a similar definition for $M_n^p$.
Then $P(m_n^p < r) = P(M_n^{1-p} > -r)$.