Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete

, Volume 32, Issue 1, pp 111–131

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

Authors

  • J. Komlós
    • Mathematical Institute of the Hungarian Academy of Sciences
  • P. Major
    • Mathematical Institute of the Hungarian Academy of Sciences
  • G. Tusnády
    • Mathematical Institute of the Hungarian Academy of Sciences
Article

DOI: 10.1007/BF00533093

Cite this article as:
Komlós, J., Major, P. & Tusnády, G. Z. Wahrscheinlichkeitstheorie verw Gebiete (1975) 32: 111. doi:10.1007/BF00533093

Summary

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 [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 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.

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© Springer-Verlag 1975