REFINEMENTS OF KOLMOGOROVS LAW OF THE ITERATED LOGARITHM

Let {S(n)} be the partial sums of a sequence {X(n)} of centred random variables. Suppose s(n)2 = ES(n)2, t(n)2 = 2 log log s(n)2 and s(n) --> infinity. It is shown that the law of the iterated logarithm (LIL) holds when t(n)X(n)/s(n) -- 0 almost surely and t(n)\X(n)\/s(n) less-than-or-equal-to Y for all n greater-than-or-equal-to 1 and some L2-integrable Y, even though it may fail if only one of the conditions holds. Moreover, when t(n)X(n)/s(n) --> 0 a.s. and EX(n)2/s(n)2 --> 0, the Central Limit Theorem implies the LIL, but the converse is not always true.