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Moment conditions in strong laws of large numbers for multiple sums and random measures

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  • Klesov, Oleg
  • Molchanov, Ilya

Abstract

The validity of the strong law of large numbers for multiple sums Sn of independent identically distributed random variables Zk, k≤n, with r-dimensional indices is equivalent to the integrability of |Z|(log+|Z|)r−1, where Z is the generic summand. We consider the strong law of large numbers for more general normalizations, without assuming that the summands Zk are identically distributed, and prove a multiple sum generalization of the Brunk–Prohorov strong law of large numbers. In the case of identical finite moments of order 2q with integer q≥1, we show that the strong law of large numbers holds with the normalization (n1⋯nr)1∕2(logn1⋯lognr)1∕(2q)+ε for any ε>0.

Suggested Citation

  • Klesov, Oleg & Molchanov, Ilya, 2017. "Moment conditions in strong laws of large numbers for multiple sums and random measures," Statistics & Probability Letters, Elsevier, vol. 131(C), pages 56-63.
  • Handle: RePEc:eee:stapro:v:131:y:2017:i:c:p:56-63
    DOI: 10.1016/j.spl.2017.08.007
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    References listed on IDEAS

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    1. Son, Ta Cong & Thang, Dang Hung, 2013. "The Brunk–Prokhorov strong law of large numbers for fields of martingale differences taking values in a Banach space," Statistics & Probability Letters, Elsevier, vol. 83(8), pages 1901-1910.
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