Suppose $(X_n)_{n=1}^\infty$ is a sequence of independent r.v's such that $P(X_n =1)=P(X_n =-1)=1/2$.
I am trying to show that $\sqrt{\frac{3}{n^3}}\sum_{i=1}^niX_i$ converges in distribution to the standard normal random variable. I am not sure how to proceed with this, do I use characteristic functions or go with the definition of convergence?