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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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