On the Exit Time of a Random Walk with Positive Drift

We study a random walk with positive drift in the first quadrant of the plane. For a given connected region $\mathcal{C}$ of the first quadrant, we analyze the number of paths contained in $\mathcal{C}$ and the first exit time from $\mathcal{C}$. In our case, region $\mathcal{C}$ is bounded by two crossing lines. It is noted that such a walk is equivalent to a path in a tree from the root to a leaf not exceeding a given height. If this tree is the parsing tree of the Tunstall or Khodak variable-to-fixed code, then the exit time of the underlying random walk corresponds to the phrase length not exceeding a given length. We derive precise asymptotics of the number of paths and the asymptotic distribution of the exit time. Even for such a simple walk, the analysis turns out to be quite sophisticated and it involves Mellin transforms, Tauberian theorems, and infinite number of saddle points.