ABSTRACT
This chapter discusses an important aspect of classical probability theory, asymptotics. It presents two of the most important results in asymptotic theory; the Slutsky and Cramer theorems, and their extensions. The chapter describes the asymptotic behavior of a sequence of real-valued random variables and its partial sums, as well as the limiting behavior of the sequence of distributions. It presents some essential results that connect probability distributions, convolutions and weak convergence. Having acquired the rigorous probabilistic framework, the chapter focuses on convergence of random sequences in probability, in distribution and a.s. convergence, in a rigorous framework. It explains how one can relax the independence assumption of the random sequence and still obtain a central limit theorem. Random series appear almost exclusively in statistics, in particular, in the form of point estimators of parameters. Therefore, it is important to consider the distribution of a partial sum and study its behavior as the sample size increases.
