UA MATH563 The central limit theorem of mathematical foundations of probability theory 25 The continuity theorem of characteristic functions of random variables

Continuity Theorem
assumes $\{{\mu }_n\},\mu$ is the probability measure,$\left\{f_n\right\},\varphi$ is their characteristic function:

1. ${\mu }_n\Rightarrow \mu$， 则${\varphi }_n\to \varphi$
2. ${\varphi }_n\to \psi$ and$\psi$ is continuous at zero, then$\psi$ is the characteristic function, ifμ n ⇒ ν \mu_n \Rightarrow \nuis also known${\mu }_n\Rightarrow \nu$， 则$\psi$是ν \ nowCharacteristic function of$\nu$ .

Explanation
According to the two conclusions of this theorem, we can get ${\varphi }_n\to \varphi$等价 于${\mu }_n\Rightarrow \mu$ . There is an interesting condition in the second conclusion,$\psi$ is continuous at 0. Why is this condition needed? We can use an example to illustrate that if this condition does not hold, the theorem does not hold:

If $X_n\sim N(0,n)$ , then$X_n$The density function of will become flatter and flatter, consider the characteristic function
$${\varphi }_{X_n}(t)=e^{-n^2{t}^2/2}\to \psi \left(t\right)=\{\begin{array}{l}1,t=0\\ 0,t\ne 0\end{array}$$

Obviously $\psi$ is not continuous at 0. It is not difficult to verify that
$${\mu }_n(-\infty ,x]={\int }_{-\infty }^{x/\sqrt{n}}\frac{1}{\sqrt{2\pi }}{e}^{-s^2/2}ds\to 1/2$$

So obviously $X_n$The limit distribution of does not exist.

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Proving the idea The
first conclusion, if ${\mu }_n\Rightarrow \mu$ , according to the monotonic convergence theorem,${\varphi }_n\to F$ ?

The second conclusion: suppose ${\varphi }_n\to \psi$ and$\psi \; is$ at$0$ is continuous, and${\mu }_n\Rightarrow \nu$ , the proof is divided into the following steps:

1. Description $\left\{{\mu }_n\right\}$ Is tight;
2. According to the Bolzano-Weierstrass theorem, $\left\{{\mu }_n\right\}$ 'S sub-column$\left\{f_{n_k}\right\}$ Weak convergence, according to the first conclusion,${\varphi }_{n_k}\to \psi$且$\psi$ is the characteristic function

Here is a proof of Durrett: