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

• D. W. Müller
Article

## Summary

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

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