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Measure, Probability, and Mathematical Finance: A Problem-Oriented Approach by Hong Xie, Chaoqun Ma, Guojun Gan

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CHAPTER 20

CENTRAL LIMIT THEOREMS

The central limit theorem is a very important result in probability theory. It states that the mean of a sufficiently large number of independent random variables will be approximately normally distributed under certain conditions. The central limit theorem has several versions. In this chapter, we present some central limit theorems.

20.1 Basic Concepts and Facts

Definition 20.1 (Degenerate Distribution). A distribution function is said to be degenerate if it is equal to Δx0 for some x0, where Δx0 is defined as

(20.1) equation

Definition 20.2 (Weak Convergence). A sequence {Fn}n≥1 of distribution functions is said to converge weakly to a distribution function F, written as Fn F or Fn(x) F(x), if

equation

where (F) = {x R : F is continuous at x}.

Definition 20.3 (Convergence in Distribution). Let {Xn}n≥1 be a sequence of random variables. The sequence is said to converge ...

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