A weighted central limit theorem

Abstract

Let X = { X , X n , n ⩾ 1 } be a sequence of centered square integrable iid random variables. Let w ̲ = { w k , k ⩾ 1 } be a sequence of reals and note S n = w 1 X 1 + ⋯ + w n X n , a n = w 1 2 + ⋯ + w n 2 . We obtain sharp sufficient conditions for the CLT of normalized weighted sums S n / a n . We also deduce a sharp version of the ASCLT.