DETECTING DEFECTS IN PERIODIC SCENERY BY RANDOM-WALKS ON Z

Thinking of a deterministic function s:Z-->N as ''scenery'' on the int egers, a random walk (Z(0), Z(1), Z(2),...) on Z generates a random re cord of scenery ''observed'' along the walk: s(Z)=(s(Z(0)), s(Z(1)),.. .). Suppose t:Z-->N is another scenery on the integers that is neither a translate of s nor a translate of the reflection of s. It has been conjectured that, under these circumstances, with a simple symmetric w alk Z the distributions of s(Z) and t(Z) are orthogonal. The conjectur e is generally known to hold for periodic s and t. In this paper we sh ow that the conjecture continues to hold for periodic sceneries that h ave been altered at finitely many locations with any symmetric walk wh ose steps are restricted to {-1, 0, +1}. If both sceneries are purely periodic and the walk is asymmetric (with steps restricted to {-1, 0, +1}), we get a somewhat stronger result. (C) 1996 John Wiley & Sons, I nc.