Journal of Applied Mathematics
Volume 2008 (2008), Article ID 639145, 10 pages
Research Article

A Strong Limit Theorem for Functions of Continuous Random Variables and an Extension of the Shannon-McMillan Theorem

Gaorong Li,1,2 Shuang Chen,3 and Sanying Feng4

1School of Finance and Statistics, East China Normal University, Shanghai 200241, China
2College of Applied Sciences, Beijing University of Technology, Beijing 100022, China
3School of Sciences, Hebei University of Technology, Tianjin 300130, China
4College of Mathematics and Science, Luoyang Normal University, Henan 471022, China

Received 1 November 2007; Revised 10 January 2008; Accepted 29 March 2008

Academic Editor: Onno Boxma

Copyright © 2008 Gaorong Li et al. 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.


By means of the notion of likelihood ratio, the limit properties of the sequences of arbitrary-dependent continuous random variables are studied, and a kind of strong limit theorems represented by inequalities with random bounds for functions of continuous random variables is established. The Shannon-McMillan theorem is extended to the case of arbitrary continuous information sources. In the proof, an analytic technique, the tools of Laplace transform, and moment generating functions to study the strong limit theorems are applied.