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

  title={An approximation of partial sums of independent RV'-s, and the sample DF. I},
  author={John Komlos and Péter Major and G{\'a}bor E. Tusn{\'a}dy},
  journal={Zeitschrift f{\"u}r Wahrscheinlichkeitstheorie und Verwandte Gebiete},
SummaryLet 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 [14] 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 [1] indicates that even if X1 has a finite moment generating function, the best possible bound for any… 
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Studies In The Theory Of Random Processes