A Chung type law of the iterated logarithm for subsequences of a Wiener process

Let {W(t), t ⩾ 0} be a standard Wiener process and {tn, n ⩾ 1} be an increasing sequence of positive numbers with tn → ∞. We consider the limit inf for the maximum of a subsequence |W(ti)|. It is proved in this paper that the Chung law of the iterated logarithm holds, i.e., lim infn→∞(tnlog log tn)−12 maxi〈n |W(ti)| = π√8 a.s. if tn − tn−1 = o(tnlog log tn) and that the assumption tn − tn−1 = o(tnlog log tn) cannot be weakened to tn − tn−1 = O(tnlog log tn).