The Annals of Applied Probability

Reconstruction of a multidimensional scenery with a branching random walk

Heinrich Matzinger, Angelica Pachon, and Serguei Popov

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We consider a $d$-dimensional scenery seen along a simple symmetric branching random walk, where at each time each particle gives the color record it observes. We show that up to equivalence the scenery can be reconstructed a.s. from the color record of all particles. To do so, we assume that the scenery has at least $2d+1$ colors which are i.i.d. with uniform probability. This is an improvement in comparison to Popov and Pachon [Stochastics 83 (2011) 107–116], where at each time the particles needed to see a window around their current position, and in Löwe and Matzinger [Ann. Appl. Probab. 12 (2002) 1322–1347], where the reconstruction is done for $d=2$ with a single particle instead of a branching random walk, but millions of colors are necessary.

Article information

Ann. Appl. Probab., Volume 27, Number 2 (2017), 651-685.

Received: March 2014
Revised: June 2015
First available in Project Euclid: 26 May 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J05: Discrete-time Markov processes on general state spaces 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)

Random walk branching random walk reconstruction algorithm


Matzinger, Heinrich; Pachon, Angelica; Popov, Serguei. Reconstruction of a multidimensional scenery with a branching random walk. Ann. Appl. Probab. 27 (2017), no. 2, 651--685. doi:10.1214/16-AAP1183.

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