ABSTRACT
In this chapter, the authors consider only linear functions of random variables X1 ,Xn,…,Xn, and prove some useful probabilistic results concerning these linear functions. Two important series of results of probability theory that play a dominant role in statistics are the laws of large numbers and limit theorems. Before the authors start our detailed discussion of the laws of large numbers and the limit theorems, they first define a few types of convergence that are useful in the probability theory. The use of characteristic functions has proven to be a very powerful tool in proving these limit theorems. Furthermore, there is a one-to-one correspondence between convergence in distribution and convergence in terms of characteristic function. Every sequence of distribution functions is weakly compact.
