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Topic: Central Limit Theorem hypotheses: which moments need to be finite?
Replies: 2   Last Post: Mar 26, 2013 9:25 AM

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andrea.panizza75@gmail.com

Posts: 6
Registered: 3/12/08
Central Limit Theorem hypotheses: which moments need to be finite?
Posted: Mar 22, 2013 4:30 AM
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Hi, all!

On the Internet and in textbooks, most of the statements of the Central Limit Theorem I found (for example here):

http://mathworld.wolfram.com/CentralLimitTheorem.html

assume the following three hypotheses:

1. sequence of real random variables X_i independent and identically distributed
2. finite mean mu
3. finite variance sigma^2

However, this one:

http://cermics.enpc.fr/~bl/Halmstad/monte-carlo/lecture-1.pdf

only assumes 1. and

2'. finite second order moment E[X_1^2]

Does 2'. implies 2. and 3.? It seems to me that it does, if the X_i is a continuous random variable. In that case,

mu=int[-inf,+inf](x*f(x)dx)= int[-inf,-1](x*f(x)dx) + int[-1,1](x*f(x)dx) +
int[1,+inf](x*f(x)dx)

The first and last terms are both less than int[-inf,inf](x^2*f(x)dx) which is finite by hypothesis. The second therm is easily bounded:

int[-1,1](x*f(x)dx) =< int[-1,1](f(x)dx) =< int[-inf,+inf](f(x)dx)=1

So mu is finite, i.e., 2. holds. Since sigma^2 = E[X_1^2] - mu^2, also 3. holds.

Does it seem correct to you? I have no idea how to extend it to real random variables which are not continuous, though. Some help here? Thanks,

Best Regards

deltaquattro



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