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 𝔼X 2 < ∞ and 𝔼X 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(·).