An extension of almost sure central limit theory

We show that if Xn are i.i.d. random variables with EX1=0,EX12=1 and (dk) is a positive numerical sequence obeying a condition similar to Kolmogorov's condition for the LIL, then letting Dn=∑k=1ndk we have1DN∑k=1NdkI{Sk/k⩽x}→12π∫-∞xe-t2/2dt.This shows that logarithmic means, used traditionally in a.s. central limit theory, can be replaced by other means lying much closer to ordinary (Cesàro) averages, leading to considerably sharper results. Our results remain valid (under suitable regularity conditions) for independent random variables Xn satisfying the weak limit theorem an-1Sn-bn⟶LH with an arbitrary distribution function H.