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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:
 [1] Baayen, P.C. and Hedrlin, Z. : The existence of well distributed sequences in compact spaces , Indag. Math. 27 (1965), 221-228. · Zbl 0166.05601 [2] Cigler, J. : Der individuelle Ergodensatz in der Theorie der Gleichverteilung mod 1 , J. reine angew. Math. 205 (1960), 91-100. · Zbl 0131.29301 · doi:10.1515/crll.1960.205.91 · resolver.sub.uni-goettingen.de · eudml:150454 [3] Cigler, J. : A characterization of well-distributed sequences , Compositio Math. 17 (1966), 263-267. · Zbl 0139.27801 · numdam:CM_1965-1966__17__263_0 · eudml:88919 [4] Helmberg, G. and Paalman-De Miranda, A. : Almost no sequence is well distributed , Indag. Math. 26 (1964), 488-492. · Zbl 0125.02704 [5] Hlawka, E. : Folgen auf kompakten Räumen , Abh. math. Sem. Univ. Hamburg 20 (1956), 233-241. · Zbl 0072.05701 · doi:10.1007/BF03374560 [6] Kemperman, J.H.B. : Probability methods in the theory of distributions modulo one , Compositio Math. 16 (1964), 106-137. · Zbl 0201.37701 · numdam:CM_1964__16__106_0 · eudml:88884 [7] Kuipers, L. and Niederreiter, H. : Uniform Distribution of Sequences , John Wiley & Sons, New York, 1974. · Zbl 0281.10001 [8] Niederreiter, H. : On the existence of uniformly distributed sequences in compact spaces , Compositio Math. 25 (1972), 93-99. · Zbl 0239.10019 · numdam:CM_1972__25_1_93_0 · eudml:89133 [9] Sun, Y. : Isomorphisms for convergence structures , National University of Singapore, 1991.
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