## Michael Drmota ; Wojciech Szpankowski - On the Exit Time of a Random Walk with Positive Drift

dmtcs:3525 - Discrete Mathematics & Theoretical Computer Science, January 1, 2007, DMTCS Proceedings vol. AH, 2007 Conference on Analysis of Algorithms (AofA 07) - https://doi.org/10.46298/dmtcs.3525
On the Exit Time of a Random Walk with Positive DriftArticle

Authors: Michael Drmota 1; Wojciech Szpankowski 2

• 1 Institut für Geometrie
• 2 Department of Computer Science [Purdue]

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.

Volume: DMTCS Proceedings vol. AH, 2007 Conference on Analysis of Algorithms (AofA 07)
Section: Proceedings
Published on: January 1, 2007
Imported on: May 10, 2017
Keywords: Random walk in the plane,Tunstall's code,number of paths,exit time,Khodak code,Mellin transform,Tauberian theorems.,[INFO.INFO-DS] Computer Science [cs]/Data Structures and Algorithms [cs.DS],[INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM],[MATH.MATH-CO] Mathematics [math]/Combinatorics [math.CO],[INFO.INFO-CG] Computer Science [cs]/Computational Geometry [cs.CG]
Funding:
Source : OpenAIRE Graph
• Combinatorial &Probabilistic Methods for Biol Sequences; Funder: National Institutes of Health; Code: 5R01GM068959-04