The Annals of Applied Probability

Scenery reconstruction in two dimensions with many colors

Matthias Löwe and Heinrich Matzinger, III

Full-text: Open access

Abstract

Kesten has observed that the known reconstruction methods of random sceneries seem to strongly depend on the one-dimensional setting of the problem and asked whether a construction still is possible in two dimensions. In this paper we answer this question in the affirmative under the condition that the number of colors in the scenery is large enough.

Article information

Source
Ann. Appl. Probab., Volume 12, Number 4 (2002), 1322-1347.

Dates
First available in Project Euclid: 12 November 2002

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1037125865

Digital Object Identifier
doi:10.1214/aoap/1037125865

Mathematical Reviews number (MathSciNet)
MR1936595

Zentralblatt MATH identifier
1018.60100

Subjects
Primary: 28A99: None of the above, but in this section 60J15

Keywords
Scenery random walk coloring

Citation

Löwe, Matthias; Matzinger, Heinrich. Scenery reconstruction in two dimensions with many colors. Ann. Appl. Probab. 12 (2002), no. 4, 1322--1347. doi:10.1214/aoap/1037125865. https://projecteuclid.org/euclid.aoap/1037125865


Export citation

References

  • [1] BENJAMINI, I. and KESTEN, H. (1996). Distinguishing sceneries by observing the sceneries along a random walk path. J. Anal. Math. 69 97-135.
  • [2] DEN HOLLANDER, W. Th. F. and KEANE, M. (1986). Ergodic properties of color records. Phy s. A 138 183-193.
  • [3] DOWNHAM, D. and FOTOPOLOUS, S. (1988). The transient behaviour of the simple random walk in the plane. J. Appl. Probab. 25 58-69.
  • [4] DVORETZKY, A. and ERDÖS, P. (1951). Some problems on random walks in space. Proc. Second Berkeley Sy mp. Math. Statist. Probab. 353-367. Univ. California Press, Berkeley.
  • [5] ERDÖS, P. and TAy LOR, S. J. (1960). Some problems concerning the structure of random walk paths. Acta Math. Acad. Sci. Hungar. 11 137-162.
  • [6] HOWARD, C. D. (1996). Detecting defects in periodic scenery by random walks in Z. Random Structures Algorithms 8 59-74.
  • [7] KESTEN, H. (1998). Distinguishing and reconstructing sceneries from observations from random walk paths. In Microsurvey s in Discrete Probability (D. Aldous and J. Propp, eds.) 75-83. Amer. Math. Soc., Providence, RI.
  • [8] LINDENSTRAUSS, E. (1999). Indistinguishable sceneries. Random Structures Algorithms 14 71-86.
  • [9] MATZINGER, H. (1997). Reconstructing a 3-color scenery by observing it along a simple random walk. Random Structures Algorithms 15 196-207.
  • [10] MATZINGER, H. (1999). Reconstruction of a one dimensional scenery seen along the path of a random walk with holding. Ph.D. thesis, Cornell Univ.
  • [11] REVESZ, P. (1990). Estimates on the largest disc covered by a random walk. Ann. Probab. 18 1784-1789.
  • [12] SPITZER, F. (1964). Principles of Random Walk. Van Nostrand, London.
  • [13] TALAGRAND, M. (1996). A new look at independence. Ann. Probab. 24 1-34.