Stationary solutions of stochastic recursions describing discrete event systems

Abstract

We consider recursions of the form x/sub n+1/=/spl phi//sub n/[x/sub n/], where {/spl phi//sub n/, n/spl ges/0} is a stationary ergodic sequence of maps from a Polish space (E, /spl epsiv/) into itself, and {x/sub n/, n/spl ges/0} are random variables taking values in (E, /spl epsiv/). The question of when stationary solutions exist for such recursions, whether they are unique, and whether there is convergence to a stationary solution starting from arbitrary initial conditions is of considerable interest in discrete event system applications. Currently available techniques can only answer such questions under strong simplifying assumptions on the statistics of {/spl phi//sub n/}/sub n/, (such as Markov assumptions), or on the nature of these maps (such as monotonicity), In this paper we introduce a new technique for studying stochastic recursions without such simplifying assumptions. To do so, we weaken the solution concept: rather than constructing 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 to the recursion. The problem of existence of a stationary solution is then translated into the problem of establishing tightness of a sequence of probability distributions, and uniqueness questions can be addressed using techniques familiar from the ergodic theory of positive Markov operators on spaces of continuous functions. >