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.

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

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.