Lleis infinitament divisibles i teoremes de pas al límit de probabilitat

Abstract

[en] In probability theory, a probability distribution is infinitely divisible if it can be expressed as the probability distribution of the sum of an arbitrary number of independent and identically distributed random variables. The objective of this work is to build a solid base of knowledge that allows us to define the infinitely divisible laws and study their properties, although we will also see other límit theorems of probability that are not strictly related to it. We will thus make a brief introduction to probability and characteristic functions, we will then study the asymptotic behavior of random series and, after that, in the third chapter, we will get into the study of probability measures. In this section we will define the weak convergence and the convolutions of probability measures. The propositions and theorems of this section will accompany us throughout the rest of this work, since they will be the basis for demonstrating numerous results in the fourth chapter. In addition, these results will allow us to define the infinitely divisible laws and study their properties. Finally, we will introduce the concept of the CPoiss law of parameter $\mu$, to finish demonstrating the Levy-Khintchine theorem.