On the set of limit points of normed sums of geometrically weighted i.i.d. unbounded random variables

Deli Li, Yongcheng Qi, Andrew Rosalsky

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For a sequence of i.i.d. unbounded random variables {Yn,n ≥ 1} and a constant b gt; 1,it is shown for, that if, then, and for almost every ω ∈ ω, the set ℂ(ω) of limit points of Wn(ω) coincides with the set [l L] where l and L are the essential infimum of Y1 and the essential supremum of Y1, respectively. For the case where, examples are given wherein the limit point set ℂ is identified and it is not necessarily the interval [l L]. The current work is a follow-up to the investigation of Li, Qi, and Rosalsky (Stochastic Analysis and Applications, 2008, 28:86-102) identifying the limit point set of Wn when Y1 is bounded; the results for unbounded Y1 are structurally different from those for bounded Y1 and are thus not merely simple extensions of the bounded case.

Original languageEnglish (US)
Pages (from-to)862-883
Number of pages22
JournalStochastic Analysis and Applications
Issue number5
StatePublished - 2010

Bibliographical note

Funding Information:
Received August 26, 2009; Accepted October 7, 2009 The research of Deli Li was partially supported by a grant from the Natural Sciences and Engineering Research Council of Canada and the research of Yongcheng Qi was partially supported by NSF Grant DMS-0604176. Address correspondence to Andrew Rosalsky, Department of Statistics, University of Florida, Gainesville, FL 32611-8545, USA; E-mail: rosalsky@stat.ufl.edu


  • Almost sure convergence
  • Essential infimum
  • Essential supremum
  • Geometric weights
  • Limit points
  • Spectrum of a distribution function
  • Sums of geometrically weighted i.i.d. unbounded random variables


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