Random Processes - Basis and applications

Random models have proved their worth in many applications. The underlying reasons for these models and their usefulness are the subject of much thought and even controversy, where we can distinguish a few main lines: i) "physics" is random - this is the model of quantum mechanics, where randomness is present from the microscopic level onwards; ii) physics is deterministic at the microscopic level, but the transition to the macroscopic leads to behavior that is described by a random model - this is the case of statistical physics; iii) physics is deterministic, but the complexity of the phenomenon is such that a random model is the most effective (as in the case of a throw of the dice); iv) the uncertainties and misunderstandings that exist about a reality, itself random or deterministic, are represented randomly.

The formal framework is that of probabilized spaces, on which we define applications in the set of reals called "random variables", then collections of random variables called "random processes". These are therefore the basis of engineering in a variety of fields: physics, economics, finance, biology, etc.

This article presents the fundamentals of random processes and illustrates them with concrete examples.

The first chapter recalls the basics of probability, and defines random variables and processes, as well as their main properties, such as covariance, stationarity and spectral representation. The second chapter gives examples of fundamental random processes, such as the Poisson process and Markov chains, which are used universally. The third chapter describes transformations of random processes, such as filtering, thresholding and identification. Finally, five chapters are devoted to a specific application (radar, image, atmospheric and optical turbulence, maintenance, financial series), with the aim of illustrating previous developments, but also describing certain specific techniques.

Mathematical developments are kept to a minimum, although some reasoning is made explicit where necessary.