Abstract
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.
Highlights
Let C ⊆ R2≥0 be a bounded connected region of the first quadrant of the plane with the property that if an integer lattice point (k1, k2) = (0, 0) with non-negative integers k1, k2 is contained in C, either (k1 − 1, k2) or (k1, k2 − 1) is in C, too
Let L(C) denote the set of lattice paths starting at the origin (0, 0) with steps of the form L = (1, 0) and R = (0, 1) such that they exit region C at the last step D
We are interested in regions C that are bounded by two lines of the form ax1 + bx2 = c1 and x1 + x2 = c2
Summary
Since P (y) = pk[1] qk for some nonnegative integers k1, k2 ≥ 0, we conclude that the above summation set can be expressed, after setting v = 2V , as k1 log[2] p + k2 log[2] q ≤ V which corresponds to the first line of the boundary of region C for our walks L(C) Imposing another condition on the phrase length (path in the parsing tree), namely, that it cannot exceed, say K, the above sum becomes. After translating the above sums into a recurrence, we apply the Mellin transform and Tauberian theorem to discover that we need to handle infinite saddle points on a line (incidently, already encountered in (8)) This leads to some oscillations in the leading term for the number of paths. We prove the central limit theorem for the exit time
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