A Supplement to the Einmahl–Li Results on Two-Sided Iterated Logarithm Type Behavior for I.I.D. Random Variables

Abstract

For a sequence of i.i.d. mean 0 random variables {X, X _n ; n ≥ 1} with partial sums , necessary and/or sufficient conditions are provided for {X, X _n ; n ≥ 1} to enjoy iterated logarithm type behavior of the form almost surely where h(·) is a positive, nondecreasing function that is slowly varying at infinity. New results are {obtained for the cases X ² < ∞ and X ² = ∞. 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 is finite or infinite where Lx = log _e (e ∨ x) for x ≥ 0. Corollaries are presented for particular choices of h(·).

Keywords:

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

Mathematics Subject Classification:

• 60F15
• 60G50

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.