Some optimal conditions for the ASCLT

István Berkes*, Siegfried Hörmann

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


Let X 1, X 2, … be independent random variables with EX k= 0 and σk2:=EXk2<∞(k≥ 1) . Set S k= X 1+ ⋯ + X k and assume that sk2:=ESk2→∞ . We prove that under the Kolmogorov condition |Xn|≤Ln,Ln=o(sn/(loglogsn)1/2) we have 1logsn2∑k=1nσk+12sk2f(Sksk)→12π∫Rf(x)e-x2/2dxa.s. for any almost everywhere continuous function f: R→ R satisfying |f(x)|≤eγx2 , γ< 1 / 2 . We also show that replacing the o in (1) by O, relation (2) becomes generally false. Finally, in the case when (1) is not assumed, we give an optimal condition for (2) in terms of the remainder term in the Wiener approximation of the partial sum process {Sn,n≥1} by a Wiener process.

Original languageEnglish
Number of pages19
JournalJournal of Theoretical Probability
Early online date6 May 2023
Publication statusE-pub ahead of print - 6 May 2023


  • Almost sure central limit theorem
  • Sums of independent random variables
  • Weighted averages

ASJC Scopus subject areas

  • Mathematics(all)
  • Statistics and Probability
  • Statistics, Probability and Uncertainty


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