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

Abstract

For a sequence of nondegenerate i.i.d. bounded random variables {Y _n , n ≥ 1} defined on a probability space (Ω, ℱ, ℙ) and a constant b > 1, it is shown for that

and that for almost every ω ∈ Ω,

where S(F _{Y 1} ) and S(F _V ) are the spectrums of the distribution functions of Y ₁ and , respectively and l and L are the essential infimum of Y ₁ and the essential supremum of Y ₁, respectively. Examples are provided showing that, in general, the above two inclusions are proper.

Keywords:

• Almost sure convergence
• Essential infimum
• Essential supremum
• Geometric weights
• Infinite Bernoulli convolution
• Iterated logarithm type behavior
• Law of the iterated logarithm
• Limit points
• Spectrum of a distribution function
• Sums of geometrically weighted i.i.d. random variables

Mathematics Subject Classification:

• 60F15

Acknowledgments

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.