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Reconstruction of periodic sceneries seen along a random walk

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  • Matzinger, Heinrich
  • Lember, Jüri

Abstract

This article was motivated by a question of Kesten. Kesten asked for which random walks periodic sceneries can be reconstructed. Among others, he asked the question for random walks which at each step can move by one or two units to the right. Previously, Howard [C.D. Howard, Distinguishing certain random sceneries on via random walks, Statist. Probab. Lett. 34 (2) (1997) 123-132] proved that all periodic sceneries can be reconstructed provided they are observed along the path of a simple random walk. We prove that for a large class of random walks it is possible to reconstruct periodic sceneries. Among others, we consider the random walk that can only move by one or two units to the right at each step. We show that in this case, the scenery can be reconstructed, provided that the two unit step is less likely than the one unit step.

Suggested Citation

  • Matzinger, Heinrich & Lember, Jüri, 2006. "Reconstruction of periodic sceneries seen along a random walk," Stochastic Processes and their Applications, Elsevier, vol. 116(11), pages 1584-1599, November.
    • Handle: RePEc:eee:spapps:v:116:y:2006:i:11:p:1584-1599
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    References listed on IDEAS

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    1. Keane, M. & den Hollander, W.Th.F., 1986. "Ergodic properties of color records," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 138(1), pages 183-193.
    2. Howard, C. Douglas, 1997. "Distinguishing certain random sceneries on ##Z## via random walks," Statistics & Probability Letters, Elsevier, vol. 34(2), pages 123-132, June.
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    Cited by:

    1. Hildebrand, Martin, 2017. "A condition for distinguishing sceneries on non-abelian groups," Stochastic Processes and their Applications, Elsevier, vol. 127(7), pages 2339-2345.
    2. Gross, Renan, 2024. "Brownian motion can feel the shape of a drum," Stochastic Processes and their Applications, Elsevier, vol. 167(C).
    3. Finucane, Hilary & Tamuz, Omer & Yaari, Yariv, 2014. "Scenery reconstruction on finite abelian groups," Stochastic Processes and their Applications, Elsevier, vol. 124(8), pages 2754-2770.

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    More about this item

    Keywords

    Scenery reconstruction Random walk;

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