On the lower limits of maxima and minima of wiener process and partial sums

Let M ⁺ (t) and ?M ^? (t) be the maximum and minimum of a Wiener process on the interval (O,t). This paper gives an integral test for P(M ⁺(t)?(t)t i.o.)=0 or 1. The case of i.i.d. random variables is also treated here. If a(t)=b(t), then our result gives Chung's law of the iterated logarithm 5], while b(t)=∞ corresponds to Hirsch's theorem 9]. Finally, a converse to Chung's LIL is given.