Reconstructing a random scenery observed with random errors along a random walk path

 We show that an i.i.d. uniformly colored scenery on ℤ observed along a random walk path with bounded jumps can still be reconstructed if there are some errors in the observations. We assume the random walk is recurrent and can reach every point with positive probability. At time k, the random walker observes the color at her present location with probability 1−δ and an error Y _k with probability δ. The errors Y _k , k≥0, are assumed to be stationary and ergodic and independent of scenery and random walk. If the number of colors is strictly larger than the number of possible jumps for the random walk and δ is sufficiently small, then almost all sceneries can be almost surely reconstructed up to translations and reflections. Received: 3 February 2002 / Revised version: 15 January 2003 Published online: 28 March 2003 Mathematics Subject Classification (2000): 60K37, 60G50 Key words or phrases:Scenery reconstruction – Random walk – Coin tossing problems