I am wondering if there is a way to derive the classical CLT from the Lindeberg-Feller CLT.
I looked at the two definitions on Wikipedia. But I am still very confused on how to start.
I was wondering if anyone could provide me with a hint or direction to start.
Thanks a lot!!
Let $X_n$ be a sequence of i.i.d. random variables with mean $\mu$ and variance $\sigma^2<\infty$. Then (using notation of Wikipedia):
$$\lim_{n\to\infty}\frac{1}{s_n^2}\sum_{k=1}^nE[(X_k-\mu)^2\cdot 1_{\{|X_k-\mu|>\epsilon s_n\}}]=\lim_{n\to\infty}\frac{E[(X_1-\mu)^2\cdot 1_{\{|X_1-\mu|>\epsilon s_n\}}]}{\sigma^2}$$
Note that $(X_1-\mu)^2\cdot 1_{\{|X_1-\mu|>\epsilon s_n\}}\leq (X_1-\mu)^2$ and $E[(X_1-\mu)^2]<\infty$. We may apply dominated convergence theorem to get that
$$\lim_{n\to\infty}\frac{E[(X_1-\mu)^2\cdot 1_{\{|X_1-\mu|>\epsilon s_n\}}]}{\sigma^2}=\frac{E[\lim_{n\to\infty} (X_1-\mu)^2\cdot 1_{\{|X_1-\mu|>\epsilon s_n\}}]}{\sigma^2}=0$$
Lindeberg's condition holds so thus
$$\frac{\sum X_n-\mu}{\sqrt{n}\sigma}\stackrel{D}{\implies} N(0,1)$$