Abstract
Consider a simple random walk on ℤ with a random coloring of ℤ. Look at the sequence of the first N steps taken in the random walk, together with the colors of the visited locations. We call this the record. From the record one can deduce the coloring of the interval in ℤ that was visited, which is of size approximately \(\sqrt N \). This is called scenery reconstruction. Now suppose that an adversary may change δN entries in the record that was obtained. What can we deduce from the record about the scenery now? In this paper we show that it is likely that we can still reconstruct a large part of the scenery.
More precisely, we show that for any θ <0.5, p > 0 and ∊ > 0, there are N0 and δ0 such that if N > N0 and δ < δ0, then with probability > 1 − p the walk is such that we can reconstruct the coloring of > Nθ integers in the scenery, up to having a number of suggested reconstructions that is less than 2∊s, where s is the number of integers whose color we reconstruct.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
J. Aaronson, Relative complexity of random walks in random sceneries, Annals of Probability 40 (2012), 2460–2482.
N. Alon and J. H. Spencer, The Probabilistic Method, Wiley Series in Discrete Mathematics and Optimization, John Wiley & Sons, Hoboken, NJ, 2016.
T. Austin, Scenery entropy as an invariant of rwrs processes, https://arxiv.org/abs/1405.1468.
I. Benjamini and H. Kesten, Distinguishing sceneries by observing the scenery along a random walk path, Journal d’Analyse Mathématique 69 (1996), 97–135.
F. den Hollander and J. E. Steif, Mixing properties of the generalized t, t−1 -process, Journal d’Analyse Mathématique 72 (1997), 165–202.
F. den Hollander and J. E. Steif, Random walk in random scenery: a survey of some recent results, in Dynamics & Stochastics, Institute of Mathematical Statistics Lecture Notes—Monograph Series, Vol. 48, Institute of Mathematical Statistics, Beachwood, OH, 2006, pp. 53–65.
W. Feller, An Introduction to Probability Theory and its Applications. Vol. I, John Wiley & Sons, New York—London—Sydney, 1968.
V. Guruswami, Bridging shannon and hamming: List error-correction with optimal rate, in Proceedings of the International Congress of Mathematicians, Vol. IV, Hindustan Book Agency, New Delhi, 2010, pp. 2648–2675.
W. Hoeffding, Probability inequalities for sums of bounded random variables, Journal of the American Statistical Association 58 (1963), 13–30.
S. A. Kalikow, t, t−1transformation is not loosely Bernoulli, Annals of Mathematics 115 (1982), 393–409.
M. Keane and W. Th. F. den Hollander, Ergodic properties of color records, Physica A 138 (1986), 183–193.
E. Lindenstrauss, Indistinguishable sceneries, Random Structures & Algorithms 14 (1999), 71–86.
M. Löwe and H. Matzinger, III, Scenery reconstruction in two dimensions with many colors, Annals of Applied Probability 12 (2002), 1322–1347.
J. Lember and H. Matzinger, Information recovery from a randomly mixed up message-text, Electronic Journal of Probability 13 (2008), 396–466.
M. Löwe, H. Matzinger and F. Merkl, Reconstructing a multicolor random scenery seen along a random walk path with bounded jumps, Electronic Journal of Probability 9 (2004), 436–507.
H. Matzinger, Reconstructing a three-color scenery by observing it along a simple random walk path, Random Structures & Algorithms 15 (1999), 196–207.
H. F. Matzinger, Reconstruction of a one dimensional scenery seen along the path of a random walk with holding, Ph.D. Thesis, Cornell University, Ithaca, NY, 1999.
I. Meilijson, Mixing properties of a class of skew-products, Israel Journal of Mathematics 19 (1974), 266–270.
H. Matzinger and S. W. W. Rolles, Reconstructing a piece of scenery with polynomially many observations, Stochastic Processes and their Applications 107 (2003), 289–300.
H. Matzinger and S. W. W. Rolles, Reconstructing a random scenery observed with random errors along a random walk path, Probability Theory and Related Fields 125 (2003), 539–577.
H. Matzinger and S. W. W. Rolles, Retrieving random media, Probability Theory and Related Fields 136 (2006), 469–507.
D. S. Ornstein, An example of a Kolmogorov automorphism that is not a Bernoulli shift, Advances in Mathematics 10 (1973), 49–62.
D. J. Rudolph, Asymptotically Brownian skew products give non-loosely Bernoulli k-automorphisms, Inventiones Mathematicae 91 (1988), 105–128.
B. Weiss, The isomorphism problem in ergodic theory, Bulletin of the American Mathematical Society 78 (1972), 668–684.
Author information
Authors and Affiliations
Corresponding author
Additional information
This work is part of the author’s PhD thesis. The author acknowledges support from the ISF grant 891/15 and ERC 2020 grant HomDyn 833423.
Rights and permissions
About this article
Cite this article
Lakrec, T. Scenery reconstruction for random walk on random scenery systems. Isr. J. Math. 248, 149–200 (2022). https://doi.org/10.1007/s11856-022-2301-y
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11856-022-2301-y