On the set of limit points of normed sums of geometrically weighted I.I.D. unbounded random variables. II

Li Deli, Yongcheng Qi, Andrew Rosalsky

Research output: Contribution to journalArticlepeer-review

Abstract

Let {Yn, n ≥ 1} be a sequence of i.i.d. random variables and let l and L denote the essential infimum of Y1 and the essential supremum of Y1, respectively. The set ¢ of almost sure limit points of Wn ≡ (b-1) ∑i=1n biYi/bn+1 where b > 1 is investigated. The new findings are for the case where Y1 is unbounded and are as follows: (i) If ¢ ∩ IR ≠ Φ, then ¢ = [l, L]; (ii) If l ∈ IR, then either ¢= {∞} or ¢ = [l, ∞]; (iii) If l = -∞ and L = ∞, then either ¢ = {∞}, ¢ = {-∞, }, ¢ = [-∞, ∞] or ¢ = {-∞, ∞}. 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 ¢ is identified, respectively, for bounded Y1 as the spectrum of the distribution function of (b - 1) ∑i=1 b-iYi and for unbounded Y1 with IE(log(max{{pipe}Y1{pipe}, e})) < ∞ as [l, L].

Original languageEnglish (US)
Pages (from-to)486-502
Number of pages17
JournalStochastic Analysis and Applications
Volume29
Issue number3
DOIs
StatePublished - May 2011

Bibliographical note

Funding Information:
Received May 19, 2010; Accepted July 20, 2010 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. Address correspondence to Andrew Rosalsky, Department of Statistics, University of Florida, P.O. Box 118545, Gainesville, FL 32611-8545, USA; E-mail: [email protected]

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

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