The random functions

Unlike the usual analytical functions, which can be rigorously determined, the numerical value of a random function remains totally or partially unpredictable.

The most familiar example of random functions is undoubtedly the capture of certain signals dependent on the time variable. In this particular case, although the time variable can be fixed very precisely by periodic sampling, the amplitude of the signal will remain shrouded in an uncertainty constituting a discrete or continuous random variable.

We shall see in the course of the article that the only rational description of random functions can be borrowed from the notion of covariance, itself derived from the correlation coefficient defined in [R 210v2] . Furthermore, it will be shown that under the stationarity condition for random functions, covariance becomes an autocorrelation function. The latter can be directly transposed under analytical signatures, or indirectly in the form of numerical data deduced from an arithmetic estimate. Whichever approach is adopted, examination of the autocorrelation signature will reveal the main stochastic dependence properties of the generating random function.

The question of the autocorrelation function is therefore closely related to the general properties of stationary variables discussed in the first paragraph. Stationarity will be illustrated by the spontaneous decay of atomic nuclei and the generation of electronic noise signals.

These two phenomena, well known to physicists, successively appropriate the Poisson probability law and the normal probability law, both with stationary parameters.

The second paragraph is devoted entirely to other properties involved in the calculation of autocorrelation functions. This mainly concerns the ergodic principle involved in calculating nth-order moments. Depending on whether the moment is calculated directly, by finding the mathematical expectation of a variable, or by determining the moment using the arithmetic mean of a large number of variables, the two results frequently converge. The ergodic principle will be illustrated by two examples taken from the physics of matter and measurement analysis.

Following on from the previous sections, the third paragraph, which is specifically dedicated to autocorrelation functions, tackles the subject by studying signals assimilated to random functions. Using examples, it will be shown that autocorrelation functions...