On the maximum of a Wiener process and its location (Q1087247)

Consider a Wiener process \(\{\) W(t), \(t\geq 0\}\), let \(M(t)=\max _{0\leq s\leq t}| W(s)|\) and \(\nu\) (t) be the location of the maximum of the absolute value of W(\(\cdot)\) in [0,t] i.e. \(| W(\nu (t))| =M(t)\). We study the limit points of \((\alpha _ tM(t),\beta _ t\nu (t))\) as \(t\to \infty\) where \(\alpha _ t\) and \(\beta _ t\) are positive, decreasing normalizing constants. Moreover, a lim inf result is proved for the length of the longest flat interval of M(t).