Essential Asymptotics and Applications

Abstract

Asymptotic theory is the study of how distributions of functions of a set of random variables behave, when the number of variables becomes large. One practical context for this is statistical sampling, when the number of observations taken is large. Distributional calculations in probability are typically such that exact calculations are difficult or impossible. For example, one of the simplest possible functions of n variables is their sum, and yet in most cases, we cannot find the distribution of the sum for fixed n in an exact closed form. But the central limit theorem allows us to conclude that in some cases the sum will behave as a normally distributed random variable, when n is large. Similarly, the role of general asymptotic theory is to provide an approximate answer to exact solutions in many types of problems, and often very complicated problems. The nature of the theory is such that the approximations have remarkable unity of character, and indeed nearly unreasonable unity of character. Asymptotic theory is the single most unifying theme of probability and statistics. Particularly, in statistics, nearly every method or rule or tradition has its root in some result in asymptotic theory. No other branch of probability and statistics has such an incredibly rich body of literature, tools, and applications, in amazingly diverse areas and problems. Skills in asymptotics are nearly indispensable for a serious statistician or probabilist.