Teoremas de limite e sua ilustração computacional

Abstract

This work aims to study the central limit theorems, as well as their computational illustration. We begin by presenting the laws of large numbers as a support for later chapters, namely the central limit theorems. The work begins with the proof of Bernoulli's theorem dated 1713, in its original version, since many years later and after several transformations, in the modern era when talking about Bernoulli's theorem, reference has not been made to the original version, but to the one demonstrated. based on Chebychev inequality. In the following sections we present the application of the law of large numbers to empirical distributions and density estimation. The highest point of this work is related to the central limit theorems, in which we present some important theorems that stand out in the scope of the study of central limit theorems, starting from names such as De Moivre and Laplace, passing through Liapounov to Lindeberg and Feller. Finally, in annexes, we present the results of the program, through a computational illustration of the law of large numbers and the central limit theorems. For the computational illustration of the law of large numbers, Bernoulli's law was chosen and it was possible to verify the validity of this law beyond the influence that the p parameter of the Bernoulli distribution has on the convergence speed. In the case of the central limit theorems, Poisson, binomial and uniform distributions were used, and similarly to the laws of large numbers, it was possible to verify the convergence to the normal distribution, in addition to the influence of the distribution on the velocity in said convergence.