On the set of limit points of normed sums of geometrically weighted I.I.D. bounded random variables

Deli Li, Yongcheng Qi, Andrew Rosalsky

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For a sequence of nondegenerate i.i.d. bounded random variables {Yn, n ≥ 1} defined on a probability space (Ω, ℱ, ℙ) and a constant b > 1, it is shown for Wn = (b - 1)∑ni=1 biYi/bn+1 that and that for almost every ω ∈ Ω, the set of limit points of Wn(ω) coincides with the set S (FV) where and S(FV) are the spectrums of the distribution functions of Y1 and V ≡ (b - 1)∑i=1 b-iYi respectively and l and L are the essential infimum of Y1 and the essential supremum of Y1, respectively. Examples are provided showing that, in general, the above two inclusions are proper.

Original languageEnglish (US)
Pages (from-to)86-102
Number of pages17
JournalStochastic Analysis and Applications
Issue number1
StatePublished - Jan 2010

Bibliographical note

Funding Information:
Received February 10, 2009; Accepted May 14, 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
  • Infinite Bernoulli convolution
  • Iterated logarithm type behavior
  • Law of the iterated logarithm
  • Limit points
  • Spectrum of a distribution function
  • Sums of geometrically weighted i.i.d. random variables


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