A note on the times of first passage for `nearly right-continuous' random walks

A natural extension of a right-continuous integer-valued random walk is one which can jump to the right by one or two units. First passage times above a given fixed level then admit a tractable Laplace transform (probability generating function). Explicit expressions for the probabilities that the respective overshoots are either $0$ or $1$, according as the random walk crosses a given level for the first time either continuously or not, also obtain. An interesting non-obvious observation, which follows from the analysis, is that any such (non-degenerate) random walk will, eventually in $n\in \mathbb{N}\cup \{0\}$, always be more likely to pass over the level $n$ for the first time with overshoot zero, rather than one. Some applications are considered.