 Minimum of an upward-biased random walk

Let $S_0=0$ and let the independent and identically distributed random variables $S_n-S_{n-1}$ be $+1$ with probability $p>1/2$ and $-1$ with probability $q=1-p>0$. The strong law of large numbers gives $S_n\to +\infty$ almost surely, so the global minimum $W$ is finite and attained. For each nonnegative integer $m$, the unit-step property and translation of the starting point giveby the biased random walk hitting probability. Subtracting consecutive tails gives the displayed zero-based geometric distribution, with success parameter $1-q/p$. Inclusion of time zero is important: it ensures $-W\ge 0$. Reflecting the sign of the path yields the corresponding maximum of a downward-biased random walk.