= Geometric maximum of an upward skip-free random walk
{title2=$\mathbb P(\sup_{n\geq0}S_n=j)=(1-e^{-\lambda_0})e^{-\lambda_0j}$}
Let the increments define an <upward skip-free random walk>, and suppose $\phi(\lambda)=\mathbb E e^{\lambda X_1}$ is finite for all $\lambda\geq0$. If $\lambda_0>0$ satisfies $\phi(\lambda_0)=1$ and $\phi'(\lambda_0)>0$, 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 $k\geq0$. Subtracting consecutive tail probabilities yields the displayed <geometric distribution> on $j=0,1,\ldots$. For real thresholds use $\lceil k\rceil$ in place of $k$.
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