Abstract
Let X1, X2, … be independent random variables with EXk = 0 and σk 2: = EXk2 < ∞ (k ≥ 1). Set Sk = X1 + ⋯ + Xk and assume that sk 2: = ESk2 → ∞. We prove that under the Kolmogorov condition [Formula: see text] we have [Formula: see text] 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.