Chapter Twelve

Limit Theorems

# 12.1 Introduction/Purpose of the Chapter

The notions in the current chapter could be presented together with the previous chapter concerning types of convergence. We decided to separate this chapter and present it on its own for two reasons. First, the convergence in distributions (weak convergence) presented here is special, and it is characteristic for the probability theory. Second, the two big theorems related to convergence in distribution (the law of large numbers and the central limit theorem) are the basis of statistics and stochastic processes, and we believe they deserve to be presented in their own chapter.

The present chapter is dedicated to the following problem. Suppose we have an experiment repeating itself with minor changes in the surrounding conditions. Can we say something about the underlying parameters of the experiment? Can we gather quantities such as the average or variance and use them to describe the conditions of the experiment? The notions in this chapter are some of the most useful in probability, but they have to be presented in the last chapter since they use all the other probability concepts presented throughout the book.

# 12.2 Vignette/Historical Notes

The Law of Large Numbers was first proved by the Swiss mathematician Jacob (Jacques) Bernoulli. Bernoulli was a Swiss mathematician, the first in the Bernoulli family, a family of famous scientists of the eighteenth century. Jacob Bernoulli's most original work was ...