= First-passage law of large numbers for an upward skip-free random walk
{title2=$T_j/j\xrightarrow{\mathrm{a.s.}}1/\mu$}
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 $j\geq1$. 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 $S_n\leq n$ gives $T_j\geq j\to\infty$. 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>.
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