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On metric theorems in the theory of uniform distribution. (English) Zbl 0777.11029
In the theory of uniform distribution it is well-known that “almost-no” sequence is well-distributed. In this note a corresponding general probabilistic result is proved. Let $$\{f_ n\}^ \infty_{n=1}$$ be a sequence of independent, identically distributed random variables on a probability space $$(\Omega,{\mathfrak B},P)$$, let the common distribution of $$f_ n$$’s be not a point measure and suppose
$$\int_{\Omega}| f_ 1| dP<\infty$$. Then the set $$U$$ of all $$\omega$$’s such that $\lim_{n\to\infty} {{f_{k+1}(\omega)+ \dots+ f_{k+n}(\omega)} \over n} =\int_ \Omega f_ 1 dP$ uniformly in $$k=0,1,\dots$$ is a null set. For the case that $$\Omega$$ is the sample space of an infinite coin-tossing and $$f_ n$$ is the random variable corresponding to the output of the $$n$$th toss the set $$U$$ is characterized as the image of all well-distributed sequences $$\text{mod } 1$$ under a natural map.
Reviewer: R.F.Tichy (Graz)
##### MSC:
 11K41 Continuous, $$p$$-adic and abstract analogues 11K36 Well-distributed sequences and other variations
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##### References:
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