- Autor
- Berkes, István
- Raseta, Marko
- TitelOn The Discrepancy And Empirical Distribution Function Of \({n_k α}\)
- Datei
- Erschienen inUniform Distribution Theory
- Band10
- Erscheinungsjahr2015
- Heft1
- Seiten1-17
- LicenceCC BY
- ISSN1336-913X
- Download Statistik688
- Peer ReviewJa
- AbstractBy a classical result of Philipp (1975), for any sequence \((n_k)_{k≥1}\) of positive integers satisfying the Hadamard gap condition, the discrepancy of \((n_k x)_{1≤k≤N}\) mod 1 satisfies the law of the iterated logarithm. For sequences \((n_k)_{k≥1}\) growing subexponentially this result becomes generally false and the asymptotic behavior of the discrepancy remains unknown. In this paper we show that for randomly sampled subsequences \((n_k)_{k≥1}\) the discrepancy \(D_N\) of \((n_k x)_{1≤k≤N}\) mod 1 and its \(L^p\) version \(D_N^(p)\) not only satisfy a sharp form of the law of the iterated logarithm, but we also describe the precise asymptotic behavior of the empirical process of the sequence \((n_k x)_{1≤k≤N}\) , leading to substantially stronger consequences.
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