TY - JOUR
A2 - Lopez-Herrero, MJ
AU - El-Taha, Muhammad
PY - 2016
DA - 2016/09/05
TI - Asymptotic Time Averages and Frequency Distributions
SP - 2741214
VL - 2016
AB - Consider an arbitrary nonnegative deterministic process (in a stochastic setting {X(t), t≥0} is a fixed realization, i.e., sample-path of the underlying stochastic process) with state space S=(-∞,∞). 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.
SN - 2090-3332
UR - https://doi.org/10.1155/2016/2741214
DO - 10.1155/2016/2741214
JF - International Journal of Stochastic Analysis
PB - Hindawi Publishing Corporation
KW -
ER -