## Abstract

Let {Y_{n}, n ≥ 1} be a sequence of i.i.d. random variables and let l and L denote the essential infimum of Y_{1} and the essential supremum of Y_{1}, respectively. The set ¢ of almost sure limit points of W_{n} ≡ (b-1) ∑_{i=1}^{n} b^{i}Y_{i}/b^{n+1} where b > 1 is investigated. The new findings are for the case where Y_{1} 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 Y_{1} as the spectrum of the distribution function of (b - 1) ∑_{i=1}^{∞} b^{-i}Y_{i} and for unbounded Y_{1} with IE(log(max{{pipe}Y_{1}{pipe}, e})) < ∞ as [l, L].

Original language | English (US) |
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Pages (from-to) | 486-502 |

Number of pages | 17 |

Journal | Stochastic Analysis and Applications |

Volume | 29 |

Issue number | 3 |

DOIs | |

State | Published - 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: 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