Essential Asymptotics And Applications
PROBABILITY FOR STATISTICS AND MACHINE LEARNING: FUNDAMENTALS AND ADVANCED TOPICS(2010)
摘要
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.
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