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

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Abstract

We study the extremal behavior of the stationary processes ξ(t) =, V(t)-t and |ξ(t)|on increasing intervals [O, T] as T →∞, 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∞-risk of several nonparametric maximum likelihood estimators.

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