An approximation of partial sums of independent RV'-s, and the sample DF. I

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Summary

Let S n =X 1+X 2+⋯+X n be the sum of i.i.d.r.v.-s, EX 1=0, EX 1 2 =1, and let T n = Y 1+Y 2+⋯+Y n be the sum of independent standard normal variables. Strassen proved in [14] that if X 1 has a finite fourth moment, then there are appropriate versions of S n and T n (which, of course, are far from being independent) such that ¦S n -T n ¦=O(n 1/4(log n)1/1(log log n)1/4) with probability one. A theorem of Bártfai [1] indicates that even if X 1 has a finite moment generating function, the best possible bound for any version of S n , T n is O(log n). In this paper we introduce a new construction for the pair S n , T n , and prove that if X 1 has a finite moment generating function, and satisfies condition i) or ii) of Theorem 1, then ¦S n -T n ¦=O(log n) with probability one for the constructed S n , T n . Our method will be applicable for the approximation of sample DF., too.