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2 results

  • COMPLETE CONVERGENCE FOR ARRAYS AND THE LAW OF THE SINGLE LOGARITHM

  • Part of
  • ALLAN GUT, ULRICH STADTMÜLLER
  • Journal:
    Bulletin of the Australian Mathematical Society / Volume 96 / Issue 2 / October 2017
    Published online by Cambridge University Press:
    29 August 2017, pp. 333-344
    Print publication:
    October 2017
    • Article
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  • The present paper is devoted to complete convergence and the strong law of large numbers under moment conditions near those of the law of the single logarithm (LSL) for independent and identically distributed arrays. More precisely, we investigate limit theorems under moment conditions which are stronger than $2p$ for any $p<2$, in which case we know that there is almost sure convergence to 0, and weaker than $E\,X^{4}/(\log ^{+}|X|)^{2}<\infty$, in which case the LSL holds.