
Central Limit Theorem hypotheses: which moments need to be finite?
Posted:
Mar 22, 2013 4:30 AM


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/montecarlo/lecture1.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

