I was wondering about the relation between different versions of central limit theorems.
(1) Classical CLT (Lindeberg–Lévy CLT) for a sequence of iid random variables with finite mean and variance.
(2) Lyapunov CLT for a sequence of independent random variables, each having a finite expected value and variance, and satisfying the Lyapunov’s condition.
(3) Lindeberg CLT for a sequence of independent random variables, each having a finite expected value and variance, and satisfying the Lindeberg's condition.
In Kai Lai Chung's book, both (1) Classical CLT and (2) Lyapunov CLT can be derived from (3) Lindeberg CLT. I was wondering if (1) Classical CLT can be derived from (2) Lyapunov CLT, i.e.,
$$ \lim_{n\to\infty} \frac{1}{s_{n}^{2+\delta}} \sum_{i=1}^{n} \operatorname{E}\big[\,|X_{i} – \mu_{i}|^{2+\delta}\,\big]
= \lim_{n\to\infty} \frac{1}{(n \sigma^2)^{1+\delta/2}} n \operatorname{E}\big[\,|X – \mu|^{2+\delta}\,\big]
= 0? $$
Thanks!
Best Answer
No: in (1) one requires a finite second moment while in (2) one requires finite $2+\delta$ moments.