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 DriftConference paper
Authors: Michael Drmota 1; Wojciech Szpankowski 2
NULL##NULL
Michael Drmota;Wojciech Szpankowski
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 C of the first quadrant, we analyze the number of paths contained in C and the first exit time from C. In our case, region 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,exit time,number of paths,Tunstall's code,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