Stochastic convergence of stack filters and Boolean networks

Stack filters are a large class of nonlinear filters that include median and ranked-order filters. Usually, these filters are analyzed in terms of their invariant signals and their convergence behavior. Convergence refers to whether iterative application of a stack filter to a signal will make it converge to an invariant signal. Previous convergence results have been deterministic results for limited classes of stack filters. In these results, the “visiting strategy” for the filter kernel as it filters a signal is arbitrary, but known a priori. Here, we use a new stochastic approach, with randomly evolving visiting strategies, that applies to all stack filters. We show that, within this framework, all stack filters make all input signals converge almost surely. A result for Boolean networks follows as a corollary