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

Deli Li; Yongcheng Qi; Andrew Rosalsky

doi:10.1080/07362994.2010.503464

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

Deli Li, Yongcheng Qi, Andrew Rosalsky

Mathematics & Statistics

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1 Scopus citations

Abstract

For a sequence of i.i.d. unbounded random variables {Y_n,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 W_n(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 W_n 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 language	English (US)
Pages (from-to)	862-883
Number of pages	22
Journal	Stochastic Analysis and Applications
Volume	28
Issue number	5
DOIs	https://doi.org/10.1080/07362994.2010.503464
State	Published - 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

Keywords

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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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",

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

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Bibliographical note

Keywords

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