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

  • J. Komlós
  • P. Major
  • G. Tusnády


Let S n =X1+X2+⋯+X n be the sum of i.i.d.r.v.-s, EX1=0, EX 1 2 =1, and let T n = Y1+Y2+⋯+Y n be 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 S n and T n (which, of course, are far from being independent) such that ¦S n -T n ¦=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 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 X1 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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bártfai, P.: Die Bestimmung der zu einem wiederkehrenden Proze\ gehörenden Verteilungsfunktion aus den mit Fehlern behafteten Daten einer einzigen Realisation. Studia Sci. Math. Hungar. 1, 161–168 (1966)Google Scholar
  2. 2.
    Bártfai, P.: über die Entfernung der Irrfahrtswege. Studia Sci. Math. Hungar. 5, 41–49 (1970)Google Scholar
  3. 3.
    Breiman, L.: Probability. Reading, Mass: Addison-Wesley, 1968Google Scholar
  4. 4.
    Brillinger, D. R.: An asymptotic representation of the sample df. Bull. Amer. Math. Soc. 75, 545–547 (1969)Google Scholar
  5. 5.
    CsörgŐ, M., Révész, P.: A new method to prove Strassen type laws of Invariance Principle I.-II. Z. Wahrscheinlichkeitstheorie verw. Geb. 31, 255–259 and 261–269 (1975)Google Scholar
  6. 6.
    ErdŐs, P., Rényi, A.: On a new law of large numbers. J. Analyse Math. 23, 103–111 (1970)Google Scholar
  7. 7.
    Kiefer, J.: On the deviations in the Skorohod-Strassen approximation scheme. Z. Wahrscheinlichkeitstheorie verw. Geb. 13, 321–332 (1969)Google Scholar
  8. 8.
    Kiefer, J.: Skorohod embedding of multivariate RV's, and the sample DF. Z. Wahrscheinlichkeitstheorie verw. Geb. 24, 1–35 (1972)Google Scholar
  9. 9.
    Petrov, V.V.: On large deviations of sums of random variables. (Russian) Vestnik Leningrad. Univ. 16, 25–37 (1961)Google Scholar
  10. 10.
    Petrov, V. V.: On sums of independent random variables. (Russian) Moscow, Nauka. 1972Google Scholar
  11. 11.
    Richter, W.: Local limit theorems for large deviations (Russian) Teor. Verojatnost. i Primenen 2, 214–229(1957)Google Scholar
  12. 12.
    Sawyer, S.: Rates of convergence for some functionals in probability. Ann. Math. Statist. 43, 273–284(1972)Google Scholar
  13. 13.
    Skorohod, A.V.: Studies in the theory of random processes. Reading, Mass: Addison-Wesley 1965Google Scholar
  14. 14.
    Strassen, V.: Almost sure behaviour of sums of independent random variables and martingales. Proc. 5th Berkeley Sympos. Math. Statist. Probab. (1965), Vol. II (part 1), 315–343. Berkeley: Univ. of Calif. Press 1967Google Scholar

Copyright information

© Springer-Verlag 1975

Authors and Affiliations

  • J. Komlós
    • 1
  • P. Major
    • 1
  • G. Tusnády
    • 1
  1. 1.Mathematical Institute of the Hungarian Academy of SciencesBudapestHungary

Personalised recommendations