Wiener过程的最大值及其定位

The Maximum of a Wiener Process and Its Location

设{W(t),t>0}是标准Wiener过程,M(t)=max|W(s)|,v(t)是M(t)的定位,即|W(v(t))|=M(t),本文证明了((1/t)v(t),(M(t))/(2tloglogt^(1/2)))的极限点集(t→∞)以概率1是K={(x,y),0≤x≤1, 0≤y≤1,x≥y^2}.

Let {W(t),t≥0} be a standard Wiener process, M(t)=max |W(s)|, and let v(t) be the location of M(t) i.e. |W(v(t))| =M(t). In this paper, it is proved that the set of limit points of (1/tv(t), ) (t→∞) fs K = {(x,y) :0≤x≤1,0≤y≤1,x≥y2} a.s..