How long does it take to see a flat Brownian path on the average?

Let Wt be a standard Brownian motion and define R(t, 1) = maxt-1[less-than-or-equals, slant]s[less-than-or-equals, slant]tWs-mint-1[less-than-or-equals, slant]s[less-than-or-equals, slant]t Ws for t[less-than-or-equals, slant]1. Given [var epsilon]>0, let [tau]([var epsilon])=min{t[greater-or-equal, slanted]1: R(t, 1)[less-than-or-equals, slant] [var epsilon]}. We prove that . We also give the lim inf behavior of R(t,1) and inf1[less-than-or-equals, slant]s[less-than-or-equals, slant]tR(s, 1).Brownian motion range of Brownian motion waiting time strong limit theorems