Stationary solutions of stochastic recursions describing discrete event systems
Recursions of the form \(x_{n+1}=\varphi _n[x_n]\), where \(\{\varphi_n, n\geq 0\}\) is a stationary ergodic sequence of maps from a Polish space (\(E,\mathcal E\)) into itself and \(\{x_n, n\geq 0\}\) are random variables taking values in (\(E,\mathcal E\)), often describe discrete event systems subject to random influences. In recursions of practical interest, the stability conditions are usually derived and existence, uniqueness and convergence are proved only under simplifying assumptions about \(\{\varphi _n\}_n\), or on the nature of these maps. As these assumptions are typically not met in practice, a solution concept known from the theory of stochastic differential equations is proposed. A pathwise solution is replaced by the construction of a probability measure on another sample space and of families of random variables on this space whose law gives a stationary solution to the examined recursion. The existence of a stationary solution is then translated into the tightness of a sequence of probability distributions. The relations to similar approaches in problems of random dynamical systems, stationary queueing systems and communication systems are mentioned. The construction of a probability measure on the selected space is described and a proposal is presented on how tightness implies stationarity. Further, a theorem on uniqueness is given along lines familiar from the ergodic theory of positive Markov operators on spaces of continuum functions. The proposed technique is illustrated by the reinterpretation of well-known solutions in simple queueing systems and by proving uniqueness of some of them. Its ability is shown to prove existence of solutions in non-monotonic recursions of the type \(x_{n+1}=\varphi _n(x_n,\xi _n)\), where \(\{\xi _n, n\geq 0\}\) is a stationary and ergodic sequence of random variables and \(\varphi \) is a deterministic function.
scientific article; zbMATH DE number 1240017