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Weak and one-sided strong laws for random variables with infinite mean

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  • Adler, André
  • Pakes, Anthony G.

Abstract

Let aj be positive weight constants and Xj be independent non-negative random variables (j=1,2,…) and Sn(a)=∑i=1naiXi. If the Xj have the same relatively stable distribution, then under mild conditions there exist constants bn→∞ such that W¯n(a)=bn−1Sn(a)→p1, i.e., a weak law of large numbers holds. If the weights comprise a regularly varying sequence, then under some additional technical conditions, this outcome can be strengthened to a strong law if and only if the index of regular variation is −1. This paper addresses a case where the Xj are not identically distributed, but rather the tail probability P(ajXj>x) is asymptotically proportional to aj(1−F(x)), where F is a relatively stable distribution function. Here the weak law holds but the strong law does not: under typical conditions almost surely lim infn→∞W¯n(a)=1 and lim supn→∞W¯n(a)=∞.

Suggested Citation

  • Adler, André & Pakes, Anthony G., 2018. "Weak and one-sided strong laws for random variables with infinite mean," Statistics & Probability Letters, Elsevier, vol. 142(C), pages 8-16.
    • Handle: RePEc:eee:stapro:v:142:y:2018:i:c:p:8-16
    DOI: 10.1016/j.spl.2018.06.008
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    References listed on IDEAS

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    1. Adler, André & Wittmann, Rainer, 1994. "Stability of sums of independent random variables," Stochastic Processes and their Applications, Elsevier, vol. 52(1), pages 179-182, August.
    2. Nakata, Toshio, 2016. "Weak laws of large numbers for weighted independent random variables with infinite mean," Statistics & Probability Letters, Elsevier, vol. 109(C), pages 124-129.
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