Let $X_n$ be a sequence of independent real RVs with mean $0$ and variance $1$. We also assume there exists $0<C$ such that $\mathbb E(X_n^3)<C$. I want to prove C.L.T.
all $X_n$ are in $\mathcal L^3(\Omega)$ so I already proved that:
$\mathbb E(Z_n=(1/\sqrt(n)\sum_{k\leq n}X_n)=0 $ and $var(Z_n)=1$ from the 1st and 2nd moments.
I can also prove that $\mathbb E(Z_n^3)\rightarrow 0$.
How do I go on?