Ergodic theorems for iterated function systems controlled by regenerative sequences
In the paper under review are considered special iterated function systems (IFS) on Polish spaces. Namely, if \((X,d)\) is a Polish space, \(S\) a measurable space and \(w:X\times S\to X\) a measurable function such that for any \(s\in S\), \(w_s\) is a continuous mapping, then the family \((X,w_s)_{s\in S}\) is called an IFS on \(X\). A stochastic sequence \((I_n)_{n\in\mathbb{N}}\) in \(S\) defines a stochastic dynamical system \(Z_n\) such that for any \(x_0\in X\), \(Z_0(x_0)=x_0\), and \[ Z_n(x_0)=w_{I_{n-1}}\circ w_{I_{n-2}}\circ\cdots\circ w_{I_0}. \] This type of stochastic dynamical system is called an IFS controlled by the stochastic sequence \((I_n)\). In the present paper are investigated IFS controlled by a regenerative process with discrete time and arbitrary state space. Mainly, the authors establish ergodic theorems of distributional and large numbers types which are uniform with respect to initial points chosen in compact sets. The asymptotic behaviour for the distribution of the random vector \((Z_n(x_0),I_n)\) is also determined.
scientific article; zbMATH DE number 1208773