Electronic Journal of Probability

Reconstructing a Multicolor Random Scenery seen along a Random Walk Path with Bounded Jumps

Matthias Loewe, Heinrich Matzinger, and Franz Merkl

Full-text: Open access

Abstract

Kesten noticed that the scenery reconstruction method proposed by Matzinger in his PhD thesis relies heavily on the skip-free property of the random walk. He asked if one can still reconstruct an i.i.d. scenery seen along the path of a non-skip-free random walk. In this article, we positively answer this question. We prove that if there are enough colors and if the random walk is recurrent with at most bounded jumps, and if it can reach every integer, then one can almost surely reconstruct almost every scenery up to translations and reflections. Our reconstruction method works if there are more colors in the scenery than possible single steps for the random walk.

Article information

Source
Electron. J. Probab., Volume 9 (2004), paper no. 15, 436-507.

Dates
Accepted: 15 May 2004
First available in Project Euclid: 6 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1465229701

Digital Object Identifier
doi:10.1214/EJP.v9-206

Mathematical Reviews number (MathSciNet)
MR2080606

Zentralblatt MATH identifier
1064.60201

Subjects
Primary: 60K37: Processes in random environments
Secondary: 60G10: Stationary processes 60J75: Jump processes

Keywords
Scenery reconstruction jumps stationary processes random walk ergodic theory

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

Loewe, Matthias; Matzinger, Heinrich; Merkl, Franz. Reconstructing a Multicolor Random Scenery seen along a Random Walk Path with Bounded Jumps. Electron. J. Probab. 9 (2004), paper no. 15, 436--507. doi:10.1214/EJP.v9-206. https://projecteuclid.org/euclid.ejp/1465229701


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