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].