Let Sn=X1+X2+⋯+Xnbe the sum of i.i.d.r.v.-s, EX1=0, EX12=1, and let Tn= Y1+Y2+⋯+Ynbe the sum of independent standard normal variables. Strassen proved in  that if X1 has a finite fourth moment, then there are appropriate versions of Snand Tn(which, of course, are far from being independent) such that ¦Sn -Tn¦=O(n1/4(log n)1/1(log log n)1/4) with probability one. A theorem of Bártfai  indicates that even if X1 has a finite moment generating function, the best possible bound for any version of Sn, Tnis O(log n). In this paper we introduce a new construction for the pair Sn, Tn, and prove that if X1 has a finite moment generating function, and satisfies condition i) or ii) of Theorem 1, then ¦Sn -Tn¦=O(log n) with probability one for the constructed Sn, Tn. Our method will be applicable for the approximation of sample DF., too.