STATIONARY SOLUTIONS OF STOCHASTIC RECURSIONS DESCRIBING DISCRETE-EVENT SYSTEMS

We consider recursions of the form x(n+1) = phi(n) [x(n)], where {phi( n), n greater than or equal to 0) is a stationary ergodic sequence of maps from a Polish space (E, E) into itself, and (x(n), n greater than or equal to 0) are random variables taking values in (E, E). Question s of existence and uniqueness of stationary solutions are of considera ble interest in discrete event system applications. Currently availabl e 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 simplify ing assumptions, by weakening the solution concept: instead of a pathw ise solution, we construct a probability measure on another sample spa ce and families of random variables on this space whose law gives a st ationary solution. The existence of a stationary solution is then tran slated into tightness of a sequence of probability distributions. Uniq ueness questions can be addressed using techniques familiar from the e rgodic theory of positive Markov operators.