A weighted central limit theorem

Let X={X,Xn,n⩾1}X={X,Xn,n⩾1} be a sequence of centered square integrable iid random variables. Let w̲={wk,k⩾1} be a sequence of reals and note Sn=w1X1+⋯+wnXnSn=w1X1+⋯+wnXn, an=w12+⋯+wn2. We obtain sharp sufficient conditions for the CLT of normalized weighted sums Sn/an. We also deduce a sharp version of the ASCLT.