Another intuitive proof of Central Limit Theorem

Question

if one prove that the distribution of the sample mean is symmetrical, and prove that its mean is the mean of the original population and it's variance is $\frac{\sigma^{2}}{N}$, is it possible to prove the CLT arguing that the only PDF with these properties has a normal distribution ?

Answer 1

As noted in the comments, no that's not enough. Furthermore, the sample mean need not be symmetrical if the underlying distribution of the samples isn't. However, you are coming close to another way of proving the central limit theorem: it's not enough to just show that the sample mean has the correct mean and variance; however, if you show that all moments of the (recentered and renormalized) sample mean converge to the moments of the standard normal, then you get a proof of the central limit theorem. This is known as the moment method or the method of moments, although "the method of moments" also refers to a different concept in statistics. For a proof of the CLT using the moment method, see this blog post by Terry Tao.