On the Set of Limit Points of Normed Sums of Geometrically Weighted I.I.D. Unbounded Random Variables. II

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 Y ₁ and the essential supremum of Y ₁, respectively. The set ℂ of almost sure limit points of where b > 1 is investigated. The new findings are for the case where Y ₁ is unbounded and are as follows: (i) If ℂ ∩ ℝ ≠ ∅, then ℂ = [l, L]; (ii) If l ∈ ℝ, then either ℂ = {∞} or ℂ = [l, ∞]; (iii) If l = − ∞ and L = ∞, then either ℂ = {∞}, ℂ = {− ∞}, ℂ = [− ∞, ∞], or ℂ = {− ∞, ∞}. 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 ℂ is identified, respectively, for bounded Y ₁ as the spectrum of the distribution function of and for unbounded Y ₁ with 𝔼(log (max {|Y ₁|, 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

Mathematics Subject Classification:

• 60F15

Acknowledgments

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.