Statistical Glossary
Central Limit Theorem:
The central limit theorem states that the sampling distribution of the mean approaches normality as the sample size increases, regardless of the probability distribution of the population from which the sample is drawn.
Strictly speaking, the above is true only for distributions with finite variance. If the variance is infinite (there are lots of such distributions) the sum or the mean approaches one of the so called "alpha-stable distributions" (of which the normal distribution is only a special case).