## Abstract

For a sequence of i.i.d. mean 0 random variables {X, X_{n}; n ≥ 1} with partial sums equation presented, necessary and/or sufficient conditions are provided for {X, X_{n}; n ≥ 1} to enjoy iterated logarithm type behavior of the form equation presented almost surely where h(·) is a positive, nondecreasing function that is slowly varying at infinity. New results are {obtained for the cases IEX^{2} < ∞ and IEX2 = ∞. The proofs rely heavily on recent work of Einmahl and Li (Annals of Probability (2005) 33:1601-1624) and new versions of those results are obtained under conditions couched in terms of an "integral test" involving whether equation presented is finite or infinite where Lx = log_{e}(e ∨ x) for x ≥ 0. Corollaries are presented for particular choices of h(·).

Original language | English (US) |
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Pages (from-to) | 1095-1110 |

Number of pages | 16 |

Journal | Stochastic Analysis and Applications |

Volume | 26 |

Issue number | 5 |

DOIs | |

State | Published - Sep 2008 |

### Bibliographical note

Funding Information:1Department of Mathematical Sciences, Lakehead University, Thunder Bay, Ontario, Canada 2Department of Mathematics and Statistics, University of Minnesota Duluth, Duluth, Minnesota, USA 3Department of Statistics, University of Florida, Gainesville, Florida, USA Abstract: For a sequence∑ of i.i.d. mean 0 random variables X Xn n ≥ 1 with partial sums Sn = i=1n Xi n ≥ 1, necessary and/or sufficient conditions are provided for X Xn n ≥ 1√ to enjoy iterated logarithm type behavior of the form 0 < lim supn→ Sn / nh n < almost surely where h · is a positive, nondecreasing function that is slowly varying at infinity. New results are obtained for the cases X2< and X2= . The proofs rely heavily on Received March 18, 2008; Accepted March 21, 2008 The authors are extremely grateful to Professor Uwe Einmahl for his interest in our work and for his careful reading of the manuscript. Professor Einmahl so kindly offered numerous substantial suggestions for improving the article. Our original version of Lemma 1 had pertained only to the particular functions h x = LLx p p ≥ 1 and h x = Lx r r > 0. Professor Einmahl suggested that we go and obtain a version of Lemma 1 for more general h · in the form of an “integral test.” Our new version of Lemma 1 has thus enabled us to present the very general Theorems 3 and 4 rather than the special cases (as in Corollaries 1 and 2) which we originally had presented. 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.

## Keywords

- Almost surely
- Cluster set
- Iterated logarithm type behavior
- Law of the iterated logarithm
- Slowly varying function
- Sums of i.i.d. random variables