A conditional limit law result on the location of the maximum of Brownian motion (Q1185545)

Let \(M_ t=\max_{0\leq s\leq t} B_ s\) denote the maximum of a standard Brownian motion \(B_ s\), \(s\geq 0\), starting at zero and let \(L_ t=\inf\{s: B_ s\geq M_ t\}\) denote the location point of the maximum in \([0,t]\). The authors obtain the following result about the limiting distribution of \(L_ t\) under the condition on \(M_ t\) being large, \[ \lim_{v\to\infty}P\{v\sqrt{t-L_ t}/t>z\mid M_ t>v\}=2(1- \phi(z)),\quad z\geq 0; \] where \(\phi(z)\) is a standard normal distribution.