Refinements of Kolmogorov's law of the iterated logarithm

Abstract

Let \\s{ S n \\s} be the partial sums of a sequence \\s{ X n \\s} of centred random variables. Suppose s 2 n = ES 2 n , t 2 n = 2 log log s 2 n and s n → ∞. It is shown that the law of the iterated logarithm (LIL) holds when t nX n / s n → 0 almost surely and t n \\vb X n \\vb/ s n ⩽ Y for all n ⩾ 1 and some L 2-integrable Y, even though it may fail if only one of the conditions holds. Moreover, when t nX n / s n → 0 a.s. and EX 2 n / s 2 n → 0, the Central Limit Theorem implies the LIL, but the converse is not always true.