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Abstract and Applied Analysis
Volume 2013, Article ID 503689, 13 pages
Research Article

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

Department of Mathematics, Toyo University, Kawagoe, Saitama 350-8585, Japan

Received 16 October 2012; Accepted 28 December 2012

Academic Editor: Haydar Akca

Copyright © 2013 Takeshi Yoshimoto. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


We study the structure of the ergodic limit functions determined in random ergodic theorems. When the r random parameters are shifted by the -shift transformation with , the major finding is that the (random) ergodic limit functions determined in random ergodic theorems depend essentially only on the 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 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.