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.


Generate Function Stochastic Process Probability Theory Mathematical Biology Normal Variable 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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