Some optimal conditions for the ASCLT

Let X ₁, X ₂, … be independent random variables with EX _k= 0 and σk2:=EXk2<∞(k≥ 1) . Set S _k= X ₁+ ⋯ + X _k and assume that sk2:=ESk2→∞ . We prove that under the Kolmogorov condition |Xn|≤Ln,Ln=o(sn/(loglogsn)1/2) we have 1logsn2∑k=1nσk+12sk2f(Sksk)→12π∫Rf(x)e-x2/2dxa.s. for any almost everywhere continuous function f: R→ R satisfying |f(x)|≤eγx2 , γ< 1 / 2 . We also show that replacing the o in (1) by O, relation (2) becomes generally false. Finally, in the case when (1) is not assumed, we give an optimal condition for (2) in terms of the remainder term in the Wiener approximation of the partial sum process {Sn,n≥1} by a Wiener process.