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Herman Rubin

Posts: 311
Registered: 2/4/10

Re: Central Limit Theorem hypotheses: which moments need to be finite?
Posted: Mar 22, 2013 1:02 PM

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On 2013-03-22, deltaquattro <andrea.panizza75@gmail.com> wrote:
> 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)

Your proof below extends to all probability distributions.

I suggest you study this. General measure theorentic probability
is not really more difficult than densities, and likelihood ratios
from densities can have any distribution on the positive reals
for which the expectation (integral as you use it) is 1.

> 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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Herman Rubin, Department of Statistics, Purdue University
hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558