Geometric maximum of an upward skip-free random walk ($\mathbb{P}({\sup }_{n\ge 0}{S}_n=j)=(1-e^{-{\lambda }_0})e^{-{\lambda }_{0}j}$)

Let the increments define an upward skip-free random walk, and suppose $\varphi (\lambda )=\mathbb{E}{e}^{\lambda X_1}$ is finite for all $\lambda \ge 0$. If ${\lambda }_0>0$ satisfies $\varphi ({\lambda }_0)=1$ and ${\varphi }^{\prime }({\lambda }_0)>0$, the exponential tilting at ${\lambda }_0$ gives integrable increments of positive mean. The strong law of large numbers makes every nonnegative integer level attainable under this tilted law. Upward jumps are at most one, so the value at a first level hit is exactly that level. The stopped likelihood ratio under exponential tilting therefore gives hitting probability $e^{-{\lambda }_{0}k}$ for integer $k\ge 0$. Subtracting consecutive tail probabilities yields the displayed geometric distribution on $j=0,1,\dots$. For real thresholds use $\lceil k\rceil$ in place of $k$.