Identifying several biased coins encountered by a hidden random walk

Suppose that attached to each site z ∈ Z is a coin with bias θ(z), and only finitely many of these coins have nonzero bias. Allow a simple random walker to generate observations by tossing, at each move, the coin attached to its current position. Then we can determine the biases {θ(z)}z∈Z, using only the outcomes of these coin tosses and no information about the path of the random walker, up to a shift and reflection of Z. This generalizes a result of Harris and Keane.

[1]  Elon Lindenstrauss.  Indistinguishable Sceneries , 1999, Random Struct. Algorithms.

[2]  C. Douglas Howard.  Orthogonality of Measures Induced by Random Walks with Scenery , 1996, Comb. Probab. Comput..

[3]  Charles R. Johnson,et al.  Topics in Matrix Analysis , 1991 .

[4]  Harry Kesten,et al.  Distinguishing sceneries by observing the scenery along a random walk path , 1996 .

[5]  Charles R. Johnson,et al.  Matrix analysis , 1985, Statistical Inference for Engineers and Data Scientists.

[6]  Harry Kesten,et al.  Distinguishing and reconstructing sceneries from observations along random walk paths , 1997, Microsurveys in Discrete Probability.

[7]  Robin Pemantle,et al.  A Phase Transition in Random coin Tossing , 2001 .

[8]  Heinrich Matzinger.  Reconstructing a three-color scenery by observing it along a simple random walk path , 1999 .

[9]  Heinrich Matzinger,et al.  Reconstructing a piece of scenery with polynomially many observations , 2003 .

[10]  C. Douglas Howard.  Detecting defects in periodic scenery by random walks on Z , 1996, Random Struct. Algorithms.

[11]  Heinrich Matzinger,et al.  Scenery reconstruction in two dimensions with many colors , 1999 .

[12]  P. Gács,et al.  Algorithms , 1992 .

[13]  W. Woess.  Random walks on infinite graphs and groups, by Wolfgang Woess, Cambridge Tracts , 2001 .

[14]  L. Saloff‐Coste.  RANDOM WALKS ON INFINITE GRAPHS AND GROUPS (Cambridge Tracts in Mathematics 138) , 2001 .

[15]  M. Keane,et al.  Random coin tossing , 1997 .

[16]  Heinrich Matzinger,et al.  Reconstructing a random scenery observed with random errors along a random walk path , 2003 .

[17]  Matthias Löwe,et al.  Reconstruction of sceneries with correlated colors , 1999 .