Abstract
A certain modified version of Kolmogorov’s strong law of large numbers is used for an extension of the result of C. Baxa and J. Schoi\(\beta \)engeier (2002) to a maximal set of uniformly distributed sequences in (0, 1) that strictly contains the set of all sequences having the form \((\{\alpha n\})_{n \in \mathbf{N}}\) for some irrational number \(\alpha \) and having the full \(\ell _1^{\infty }\)-measure, where \(\ell _1^{\infty }\) denotes the infinite power of the linear Lebesgue measure \(\ell _1\) in (0, 1).
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Notes
- 1.
We say that a family \((f_k)_{k \in \mathbf {N}}\) of elements of \(\mathscr {B}[0,1]\) defines a uniform convergence on [0, 1], if for each \((x_n)_{n \in N} \in [0,1]^{\infty }\) the validity of the condition \(\lim _{N \rightarrow \infty }\frac{1}{N}\sum _{n = 1}^Nf_k(x_n) = \int _{[0,1]} f_k(x)dx\) for \(k \in N\) implies that \((x_n)_{n \in N}\) is uniformly distributed on [0, 1]. Indicator functions of closed subintervals of [0, 1] with rational endpoints is an example of such a family.
References
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Pantsulaia, G. (2016). Calculation of Improper Integrals by Using Uniformly Distributed Sequences. In: Applications of Measure Theory to Statistics. Springer, Cham. https://doi.org/10.1007/978-3-319-45578-5_1
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DOI: https://doi.org/10.1007/978-3-319-45578-5_1
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