Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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!

share|improve this question

1 Answer 1

up vote 3 down vote accepted

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

share|improve this answer
    
@Didier: Thanks! Suppose 2+δ moments are finite in (1), will Lyapunov condition be satisfied? –  Tim May 13 '11 at 1:26
    
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? –  Tim May 13 '11 at 1:30
1  
Is first moment finite when the second is? Wow... –  Did May 13 '11 at 1:32
1  
Thanks! What do you mean by Wow? –  Tim May 13 '11 at 1:34
1  
@Tim: Wow. :) –  cardinal May 13 '11 at 2:44

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.