POINTS OF INCREASE FOR RANDOM-WALKS

Say that a sequence S-0,...,S-n has a (global) point of increase at k if S-k is maximal among S-0,...,S-k and minimal among S-k,...S-n. We g ive an elementary proof that an n-step symmetric random walk on the li ne has a (global) point of increase with probability comparable to 1/l og n. (No moment assumptions are needed.) This implies the classical f act, due to Dvoretzky, Erdos and Kakutani (1961), that Brownian motion has no points of increase.