Stationary solutions of stochastic recursions describing discrete event systems

Abstract

We consider recursions of the form x n + 1 = ϕ n [ x n ], where { ϕ n , n ≥ 0} is a stationary ergodic sequence of maps from a Polish space ( E, E ) into itself, and { x n , n ≥ 0} are random variables taking values in ( E, E ). Questions of existence and uniqueness of stationary solutions are of considerable interest in discrete event system applications. Currently available techniques use simplifying assumptions on the statistics of { ϕ n } n (such as Markov assumptions), or on the nature of these maps (such as monotonicity). We introduce a new technique, without such simplifying assumptions, by weakening the solution concept: instead of a pathwise solution, we construct a probability measure on another sample space and families of random variables on this space whose law gives a stationary solution. The existence of a stationary solution is then translated into tightness of a sequence of probability distributions. Uniqueness questions can be addressed using techniques familiar from the ergodic theory of positive Markov operators