Iterated logarithm type behavior for weighted sums of i.i.d. random variables

For a sequence of i.i.d. mean 0 random variables {X,Xn;n≥1} with weighted partial sums Sn(X,w(⋅))=∑k=1nw(kn)Xk,n≥1 where w(t),0≤t≤1 is a Lipschitz function of order 1 with ‖w(⋅)‖2=∫01w2(t)dt>0, necessary and sufficient conditions are provided for X to enjoy iterated logarithm type behavior of the form 0