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

Abstract

For a sequence of i.i.d. unbounded random variables {Y n , n ≥ 1} and a constant b > 1, it is shown for that if 𝔼(log (max {|Y 1|, e})) < ∞, then and for almost every ω ∈ Ω, where l and L are the essential infimum of Y 1 and the essential supremum of Y 1, respectively. For the case where 𝔼(log (max {|Y 1|, e})) = ∞, 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 Y 1 is bounded; the results for unbounded Y 1 are structurally different from those for bounded Y 1 and are thus not merely simple extensions of the bounded case.