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

Deli Li; Yongcheng Qi; Andrew Rosalsky

doi:10.1080/07362990903417987

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

Deli Li, Yongcheng Qi, Andrew Rosalsky

Mathematics & Statistics

Research output: Contribution to journal › Article › peer-review

2 Scopus citations

Abstract

For a sequence of nondegenerate i.i.d. bounded random variables {Y_n, n ≥ 1} defined on a probability space (Ω, ℱ, ℙ) and a constant b > 1, it is shown for W_n = (b - 1)∑ⁿ_i=1 bⁱY_i/bⁿ⁺¹ that and that for almost every ω ∈ Ω, the set of limit points of W_n(ω) coincides with the set S (F_V) where and S(F_V) are the spectrums of the distribution functions of Y₁ and V ≡ (b - 1)∑^∞_i=1 b^-iY_i 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 language	English (US)
Pages (from-to)	86-102
Number of pages	17
Journal	Stochastic Analysis and Applications
Volume	28
Issue number	1
DOIs	https://doi.org/10.1080/07362990903417987
State	Published - 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

Keywords

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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abstract = "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.",

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author = "Deli Li and Yongcheng Qi and Andrew Rosalsky",

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",

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N2 - 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.

AB - 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.

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KW - Spectrum of a distribution function

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

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Bibliographical note

Keywords

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