# 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 Δ_{x0} for some *x*_{0}, where Δ_{x0} is defined as

(20.1)

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

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

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

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