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Exact strong laws of large numbers for independent random fields

Volume 66 / 2018

Paweł Kurasiński, Przemysław Matuła, André Adler Bulletin Polish Acad. Sci. Math. 66 (2018), 179-188 MSC: Primary 60F15; Secondary 60G60. DOI: 10.4064/ba8153-9-2018 Published online: 19 October 2018

Abstract

Let $\{ X_{\underline{n}}, \underline{n}\in \mathbb{N}^{d}\}$ be a family of independent random variables with multidimensional indices (a random field) with the same distribution as the r.v. $X.$ A necessary and sufficient condition for the strong law of large numbers in this setting is $\mathbb E \vert X\vert \log _{+}^{d-1}\vert X\vert \lt \infty.$ Our goal is to study the almost sure convergence of normalized or weighted sums in the case when this moment condition is not satisfied.

Authors

  • Paweł KurasińskiInstitute of Mathematics
    Marie Curie-Skłodowska University
    Pl. M. Curie-Skłodowskiej 1
    20-031 Lublin, Poland
    e-mail
  • Przemysław MatułaInstitute of Mathematics
    Marie Curie-Skłodowska University
    Pl. M. Curie-Skłodowskiej 1
    20-031 Lublin, Poland
    e-mail
  • André AdlerDepartment of Mathematics
    Illinois Institute of Technology
    Chicago, IL 60616, U.S.A.
    e-mail

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