Intuitive CLT theorem

Question

What i've deduced studying some introductory material about probability is that, in layman terms , the CLT states:

the sum of independent and identically distributed random variables tends to a Gaussian Distribution

Is it correct ?

Answer 1

Not completely.

There is an extra condition on CLT: $$\mathbb EX_1^2<\infty$$or equivalently:$$\mathsf{Var}(X_1)<\infty$$

Here $X_1$ denotes one of the iid random variables that are involved.

Answer 2

Even if you want to avoid mentioning the specific conditions (that your distributions have zero mean and defined variance) which are arguably less important for a lay person, the thing that tends to a Gaussian distribution is not quite the sum, it is the sum divided by the square root of the number of summands. The plain sum of the distributions will just diverge.