Does C.L.T holds for this case? Let $X_k$ be independent variables with $\mathbb E(X_k)=0$ and $\operatorname {var}(X_k)=1$ 
Assume there exists $0<C$ for all $k$ we have $\mathbb E(|X_k|^3)<C$
[bounded 3rd moment]
Does Central Limit Theorem holds here?
If yes, where can I find a proof?
If no, any guidance to building a counter example?
 A: The conclusion of CLT here follows from the Lyapunov CLT. This is essentially the simplest generalization of the classical CLT, and it uses information about higher moments of the $X_k$. You assume that $E[X_k]=\mu_k$ exists and $\operatorname{Var}(X_k)=\sigma^2_k$ exists, and also that $E[|X_k-\mu_k|^{2+\delta}]=t_k$ exists for some $\delta>0$. You then introduce $S_n^2=\sum_{k=1}^n \sigma_k^2$ and $T_n=\sum_{k=1}^n t_k$, and you need to be able to control $T_n$ by $S_n$ in the sense that:
$$\lim_{n \to \infty} \frac{T_n}{S_n^{2+\delta}} = 0.$$
(It is easy enough to remember the exponents: they are set up so that the ratio is dimensionless.) In your case $\delta=1$, $t_k \leq C$, and $S_n^2=n$, so that it is enough to observe
$$\lim_{n \to \infty} \frac{Cn}{n^{3/2}} = 0.$$
The Lyapunov CLT is a special case of the more general Lindeberg CLT, which is essentially "the" CLT for the independent, non-identically distributed case. Lindeberg avoids higher moments by making a more detailed hypothesis about the smallness of the tails.
