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International Journal of Stochastic Analysis
Volume 2016 (2016), Article ID 2741214, 10 pages
Research Article

Asymptotic Time Averages and Frequency Distributions

Department of Mathematics and Statistics, University of Southern Maine, 96 Falmouth Street, Portland, ME 04104-9300, USA

Received 7 March 2016; Accepted 17 July 2016

Academic Editor: MJ Lopez-Herrero

Copyright © 2016 Muhammad El-Taha. 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.


Consider an arbitrary nonnegative deterministic process (in a stochastic setting is a fixed realization, i.e., sample-path of the underlying stochastic process) with state space . Using a sample-path approach, we give necessary and sufficient conditions for the long-run time average of a measurable function of process to be equal to the expectation taken with respect to the same measurable function of its long-run frequency distribution. The results are further extended to allow unrestricted parameter (time) space. Examples are provided to show that our condition is not superfluous and that it is weaker than uniform integrability. The case of discrete-time processes is also considered. The relationship to previously known sufficient conditions, usually given in stochastic settings, will also be discussed. Our approach is applied to regenerative processes and an extension of a well-known result is given. For researchers interested in sample-path analysis, our results will give them the choice to work with the time average of a process or its frequency distribution function and go back and forth between the two under a mild condition.