An Extremal Limit Theorem for the Argmax Process of Brownian Motion Minus a Parabolic Drift

AbstractWe study the extremal behavior of the stationary processes $${\xi }\left( t \right) = V\left( t \right) - t$$ and $$\left| {\xi \left( t \right)} \right|$$ , on increasing intervals [0,T], as $$T \to \infty$$ , where V(t) is the location of the maximum of standard two-sided Brownian motion minus a parabolic drift. The result can be applied to the asymptotic behavior of the $$L_\infty$$ -risk of several nonparametric maximum likelihood estimators.