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

Li Deli; Yongcheng Qi; Andrew Rosalsky

doi:10.1080/07362994.2011.564449

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

Li Deli, Yongcheng Qi, Andrew Rosalsky

Mathematics & Statistics

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

Abstract

Let {Y_n, n ≥ 1} be a sequence of i.i.d. random variables and let l and L denote the essential infimum of Y₁ and the essential supremum of Y₁, respectively. The set ¢ of almost sure limit points of W_n ≡ (b-1) ∑_i=1ⁿ bⁱY_i/bⁿ⁺¹ where b > 1 is investigated. The new findings are for the case where Y₁ is unbounded and are as follows: (i) If ¢ ∩ IR ≠ Φ, then ¢ = [l, L]; (ii) If l ∈ IR, then either ¢= {∞} or ¢ = [l, ∞]; (iii) If l = -∞ and L = ∞, then either ¢ = {∞}, ¢ = {-∞, }, ¢ = [-∞, ∞] or ¢ = {-∞, ∞}. Illustrative examples are referenced or provided showing that each of the various alternatives can hold. The current work is a continuation of the investigations of Li et al. [3, 4] wherein the set ¢ is identified, respectively, for bounded Y₁ as the spectrum of the distribution function of (b - 1) ∑_i=1^∞ b^-iY_i and for unbounded Y₁ with IE(log(max{{pipe}Y₁{pipe}, e})) < ∞ as [l, L].

Original language	English (US)
Pages (from-to)	486-502
Number of pages	17
Journal	Stochastic Analysis and Applications
Volume	29
Issue number	3
DOIs	https://doi.org/10.1080/07362994.2011.564449
State	Published - May 2011

Bibliographical note

Funding Information:
Received May 19, 2010; Accepted July 20, 2010 The research of D. L. was partially supported by a grant from the Natural Sciences and Engineering Research Council of Canada. The research of Y. Q. was partially supported by NSF Grant DMS-0604176. Address correspondence to Andrew Rosalsky, Department of Statistics, University of Florida, P.O. Box 118545, 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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title = "On the set of limit points of normed sums of geometrically weighted I.I.D. unbounded random variables. II",

abstract = "Let {Yn, n ≥ 1} be a sequence of i.i.d. random variables and let l and L denote the essential infimum of Y1 and the essential supremum of Y1, respectively. The set ¢ of almost sure limit points of Wn ≡ (b-1) ∑i=1n biYi/bn+1 where b > 1 is investigated. The new findings are for the case where Y1 is unbounded and are as follows: (i) If ¢ ∩ IR ≠ Φ, then ¢ = [l, L]; (ii) If l ∈ IR, then either ¢= {∞} or ¢ = [l, ∞]; (iii) If l = -∞ and L = ∞, then either ¢ = {∞}, ¢ = {-∞, }, ¢ = [-∞, ∞] or ¢ = {-∞, ∞}. Illustrative examples are referenced or provided showing that each of the various alternatives can hold. The current work is a continuation of the investigations of Li et al. [3, 4] wherein the set ¢ is identified, respectively, for bounded Y1 as the spectrum of the distribution function of (b - 1) ∑i=1∞ b-iYi and for unbounded Y1 with IE(log(max{{pipe}Y1{pipe}, e})) < ∞ as [l, L].",

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",

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

note = "Funding Information: Received May 19, 2010; Accepted July 20, 2010 The research of D. L. was partially supported by a grant from the Natural Sciences and Engineering Research Council of Canada. The research of Y. Q. was partially supported by NSF Grant DMS-0604176. Address correspondence to Andrew Rosalsky, Department of Statistics, University of Florida, P.O. Box 118545, Gainesville, FL 32611-8545, USA; E-mail: rosalsky@stat.ufl.edu",

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N2 - Let {Yn, n ≥ 1} be a sequence of i.i.d. random variables and let l and L denote the essential infimum of Y1 and the essential supremum of Y1, respectively. The set ¢ of almost sure limit points of Wn ≡ (b-1) ∑i=1n biYi/bn+1 where b > 1 is investigated. The new findings are for the case where Y1 is unbounded and are as follows: (i) If ¢ ∩ IR ≠ Φ, then ¢ = [l, L]; (ii) If l ∈ IR, then either ¢= {∞} or ¢ = [l, ∞]; (iii) If l = -∞ and L = ∞, then either ¢ = {∞}, ¢ = {-∞, }, ¢ = [-∞, ∞] or ¢ = {-∞, ∞}. Illustrative examples are referenced or provided showing that each of the various alternatives can hold. The current work is a continuation of the investigations of Li et al. [3, 4] wherein the set ¢ is identified, respectively, for bounded Y1 as the spectrum of the distribution function of (b - 1) ∑i=1∞ b-iYi and for unbounded Y1 with IE(log(max{{pipe}Y1{pipe}, e})) < ∞ as [l, L].

AB - Let {Yn, n ≥ 1} be a sequence of i.i.d. random variables and let l and L denote the essential infimum of Y1 and the essential supremum of Y1, respectively. The set ¢ of almost sure limit points of Wn ≡ (b-1) ∑i=1n biYi/bn+1 where b > 1 is investigated. The new findings are for the case where Y1 is unbounded and are as follows: (i) If ¢ ∩ IR ≠ Φ, then ¢ = [l, L]; (ii) If l ∈ IR, then either ¢= {∞} or ¢ = [l, ∞]; (iii) If l = -∞ and L = ∞, then either ¢ = {∞}, ¢ = {-∞, }, ¢ = [-∞, ∞] or ¢ = {-∞, ∞}. Illustrative examples are referenced or provided showing that each of the various alternatives can hold. The current work is a continuation of the investigations of Li et al. [3, 4] wherein the set ¢ is identified, respectively, for bounded Y1 as the spectrum of the distribution function of (b - 1) ∑i=1∞ b-iYi and for unbounded Y1 with IE(log(max{{pipe}Y1{pipe}, e})) < ∞ as [l, L].

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

Abstract

Bibliographical note

Keywords

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