Verteilungs-invarianzprinzipien für das starke gesetz der gro\en zahl

  • D. W. Müller


The asymptotic behaviour of the stochastic process \(k \to \frac{1}{k}\sum\limits_{i{\text{ }}\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \leqslant } {\text{ }}k} {X_i }\) as k→∞-X ={X i : i=1,2,⋯ being a sequence of independent random variables having mean 0 and positive finite variance, satisfying both Lindeberg's condition and the strong law of large numbers — is studied by means of a distribution invariance principle. This invariance principle sharpens the classical one due to Donsker and Prokhorov describing the “weak” asymptotic behaviour of partial sums of independent random variables on a semi-infinite time interval. The topology of the path space being appropriately chosen it allows to compute the limit distributions of certain functionals associated to X, such as
$$X \to \left( {\sum\limits_{i{\text{ }}\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \leqslant } {\text{ }}n} {EX_i^2 } } \right)^{1/2} \mathop {\max }\limits_{k{\text{ }}\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \geqslant } {\text{ }}n} \frac{1}{k}\left| {\sum\limits_{i{\text{ }}\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \leqslant } {\text{ }}k} {X_i } } \right|{\text{ (}}n \to \infty {\text{)}}{\text{.}}$$

Moreover, for uniformly bounded variables X i , a general estimate of the rapidity of convergence is derived and applied to various special cases


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Chung, K. L.: On the maximum partial sums of sequences of independent random variables. Trans. Amer. math. Soc. 64, 205–233 (1948).Google Scholar
  2. 2.
    Donsker, M. D.: An invariance principle for certain probability limit theorems. Mem. Amer. math. Soc. 6, 1–12 (1951).Google Scholar
  3. 3.
    Freedman, D. A.: Some invariance principles for functionals of a Markov chain. Ann. math. Statistics 38, 1–7 (1967).Google Scholar
  4. 4.
    Ito, K., and H. Mc Kean: Diffusion processes and their sample paths. Berlin-Heidelberg-New York: Springer 1965.Google Scholar
  5. 5.
    Krickeberg, K.: Wahrscheinlichkeitsoperatoren von Verteilungen in VektorrÄumen. Trans. third Prague Conf. Information Theory, statist. Decision Functions, Random Processes 1962, 441–452 (1964).Google Scholar
  6. 6.
    Müller, D. W.: Non-standard proofs of invariance principles in probability theory. (Im Erscheinen begriffen.)Google Scholar
  7. 7.
    Prokhorov, Yu. V.: Convergence of random processes and limit theorems in probability theory. Theor. Probab. Appl. 1, 157–214 (1956).Google Scholar
  8. 8.
    Skorokhod, A. V.: A limit theorem for sums of independent random variables. Soviet Math. Dokl. 1, 810–811 (1960).Google Scholar
  9. 9.
    —: A limit theorem for homogeneous Markov chains. Theor. Probab. Appl. 8, 61–70 (1963).Google Scholar
  10. 10.
    —: Studies in the theory of random processes. Reading, Mass.: Addison-Wesley 1965.Google Scholar
  11. 11.
    Strassen, V.: Almost sure behavior of sums of independent random variables and martingales. Proc. Fifth Berkeley Sympos. math. Statist. Probability, 315–343.Google Scholar

Copyright information

© Springer-Verlag 1968

Authors and Affiliations

  • D. W. Müller
    • 1
  1. 1.Erlangen-NürnbergMathematisches Institut der UniversitÄtErlangen

Personalised recommendations