On the Dependence of the Limit Functions on the Random Parameters in Random Ergodic Theorems

Abstract

We study the structure of the ergodic limit functions determined in random ergodic theorems. When the r random parameters are shifted by the $r_0$-shift transformation with $r_0\in \{1,2,\dots ,r\}$, the major finding is that the (random) ergodic limit functions determined in random ergodic theorems depend essentially only on the $r\text{—}{r}_0$ random parameters. Some of the results obtained here improve the earlier random ergodic theorems of Ryll-Nardzewski (1954), Gladysz (1956), Cairoli (1964), and Yoshimoto (1977) for positive linear contractions on $L_1$ and Woś (1982) for sub-Markovian operators. Moreover, applications of these results to nonlinear random ergodic theorems for affine operators are also included. Some examples are given for illustrating the relationship between the ergodic limit functions and the random parameters in random ergodic theorems.