A Forgetting Property of Reciprocal Chains

Reciprocal chains (RC) are a class of discrete-index, finite-state stochastic processes with a generalized Markov nearest-neighbour property. All Markov chains (MC) are RC but not conversely. It is well known that a sufficiently regular MC “forgets” its initial distribution in a geometric manner. This article addresses the issues of forgetting for RC. Using the properties of inhomogeneous products of nonnegative matrices, this article establishes that a RC forget its endpoint distributions in a more complicated manner. In particular, this article shows that the forgetting function for the initial conditions is bounded by a superposition of increasing and decreasing geometric terms. Implications for modeling using RC are then discussed. The particular case of those RC that are MC with prescribed marginal distributions on the endpoints is considered. The tightness of the forgetting bounds is illustrated with numerical examples.