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1The exact meaning of this statement is related to some refined mathematical considerations which are, in fact, closely associated with the way a random function arises, usually in an actual physical context. As already emphasized in the Introduction, in order to apply probabilistic methods, we must have an experiment which can be repeated many times under similar conditions and which can lead to different outcomes. The set Ω of all possible outcomes ω* of such an experiment (the set of so-called elementary events) plays a basic role in Kolmogorov’s axiomatic formulation of probability theory (see, e.g., Kolmogorov, 1956; Cramér, 1962; Shiryaev (1980); or any other modern advanced text on probability). A random variable X is a quantity which takes different numerical values for different outcomes of an experiment. In other words, a random variable is a numerical function of the point ω* of the set Ω. Therefore, it would be more accurate to write X*) instead of X. (This was not done anywhere above, since in probability theory, the dependence of random variables on an elementary event ω* is traditionally suppressed.) From this point of view, a random function X(t) on T should be defined as a function X(t*) of two variables t and ω*.

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© 1987 Springer-Verlag New York Inc.

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Yaglom, A.M. (1987). Chapter 1. In: Correlation Theory of Stationary and Related Random Functions. Springer Series in Statistics. Springer, New York, NY.

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  • Print ISBN: 978-1-4612-9090-2

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