Upward skip-free random walk ($X_n\in \mathbb{Z},X_n\le 1$)

An upward skip-free random walk has integer-valued increments bounded above by one. Arbitrarily large downward jumps are permitted, but a first crossing of a nonnegative integer level $k$ from below must land exactly at $k$. This removes the overshoot from the stopped likelihood ratio under exponential tilting and permits the exact geometric maximum of an upward skip-free random walk. For a real threshold $k$, the first crossing is the crossing of $\lceil k\rceil$, so the overshoot-free formula must use that integer level.

Suppose an integer-valued random walk starts at $S_0=0$, has upward increments at most one, and satisfies $S_n/n\to \mu >0$ almost surely. Write $T_j$ for its first-passage time to $j\ge 1$. On the event of almost sure convergence, the walk tends to $+\infty$ and cannot jump over any positive integer, so every $T_j$ is finite. The bound $S_n\le n$ gives $T_j\ge j\to \infty$. Along this subsequence, $j/T_j=S_{T_j}/T_j\to \mu$ on each such sample path, proving the displayed reciprocal limit. No independence between the hitting times and the path is required. The strong law of large numbers supplies the hypothesis for integrable independent and identically distributed random variables as steps with positive mean.

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 $k$.