Random variables $X_1,X_2,...$ are independent and $$P(X_k=k)=P(X_k=−k)=1/2.$$ Let $s^2_n=\sum^n_{k=1}$Var$X_k$. Is the sequence of random variables $$\frac{X_1 + X_2 + ... + X_n}{s_n}$$ converge according to any distribution, and if so, to what limit?
I think I should use Lindenberg's condition, but it seems incomprehensible for me.