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

Abstract

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)<a(t)√t M−(t)<b(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.