## Abstract

For a sequence of i.i.d. unbounded random variables {Y_{n},n ≥ 1} and a constant b gt; 1,it is shown for, that if, then, and for almost every ω ∈ ω, the set ℂ_{(ω)} of limit points of W_{n}(ω) coincides with the set [l L] where l and L are the essential infimum of Y_{1} and the essential supremum of Y_{1}, respectively. For the case where, 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.

Original language | English (US) |
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Pages (from-to) | 862-883 |

Number of pages | 22 |

Journal | Stochastic Analysis and Applications |

Volume | 28 |

Issue number | 5 |

DOIs | |

State | Published - 2010 |

### Bibliographical note

Funding Information:Received August 26, 2009; Accepted October 7, 2009 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. Address correspondence to Andrew Rosalsky, Department of Statistics, University of Florida, Gainesville, FL 32611-8545, USA; E-mail: rosalsky@stat.ufl.edu

## 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