Scenery Reconstruction on Finite Abelian Groups

We consider the question of when a random walk on a finite abelian group with a given step distribution can be used to reconstruct a binary labeling of the elements of the group, up to a shift. Matzinger and Lember (2006) give a sufficient condition for reconstructibility on cycles. While, as we show, this condition is not in general necessary, our main result is that it is necessary when the length of the cycle is prime and larger than 5, and the step distribution has only rational probabilities. We extend this result to other abelian groups.Comment: 16 pages, 2 figure

On the exchange of intersection and supremum of sigma-fields in filtering theory

We construct a stationary Markov process with trivial tail sigma-field and a nondegenerate observation process such that the corresponding nonlinear filtering process is not uniquely ergodic. This settles in the negative a conjecture of the author in the ergodic theory of nonlinear filters arising from an erroneous proof in the classic paper of H. Kunita (1971), wherein an exchange of intersection and supremum of sigma-fields is taken for granted.Comment: 20 page