Limit Theorems and Fluctuations

Abstract

Sums of random variables are a fascinating subject for they lead to certain universal types of behavior. More precisely, if we add together independent random variables distributed according to the same probability density functions and not too widely scattered in the sense that they have a finite second moment, then the new random variable obtained in this way will be described to a good approximation by a Gaussian random variable. This property has a great many applications in physics. We shall describe a certain number of them: the random walk, speckle in coherent imaging, particle diffusion, and Gaussian noise, which is a widely used model in physics. The Gaussian distribution is not the only one to appear as a limiting case. The Poisson distribution can be introduced by analogous arguments and it is also very important because it provides simple models of fluctuations resulting from detection of low particle fluxes.