Stationary solutions of stochastic recursions describing discrete event systems

We consider recursions of the form xn + 1 = [phi]n[xn], where {[phi]n, n >= 0} is a stationary ergodic sequence of maps from a Polish space (E, ) into itself, and {xn, n >= 0} are random variables taking values in (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 {[phi]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