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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!

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No: in (1) one requires a finite second moment while in (2) one requires finite $2+\delta$ moments.

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    @Didier: Thanks! Suppose 2+δ moments are finite in (1), will Lyapunov condition be satisfied?2011-05-13
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    Also why is only second or 2+δ moment is required to be finite in (1) or (2)? how about the first moment? Is first moment finite when the second is?2011-05-13
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    In (1) the random variables are i.i.d. Lyapunov condition holds for i.i.d. random variables with finite $2+\delta$ moments (but not only for these). Is this your question?2011-05-13
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    *Is first moment finite when the second is?* Wow...2011-05-13
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    Yes, why is Lyapunov condition holds for i.i.d. random variables with finite 2+δ moments?2011-05-13
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    *Lyapunov condition holds for i.i.d. random variables with finite 2+delta moments?* Yes, as you pretty much wrote it yourself.2011-05-13
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    Thanks! What do you mean by *Wow*?2011-05-13
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    @Tim: [Wow](http://math.stackexchange.com/questions/21460/how-to-show-that-lp-spaces-are-nested). :)2011-05-13