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  • Autor
    • Berkes, István
    • Raseta, Marko
  • TitelOn The Discrepancy And Empirical Distribution Function Of \({n_k α}\)
  • Volltext
  • Erschienen inUniform Distribution Theory
  • Band10
  • Erscheinungsjahr2015
  • Heft1
  • Seiten1-17
  • LicenceCC BY
  • ISSN1336-913X
  • ZugriffsrechteCC-BY
  • Download Statistik40
  • 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.