The tail of the maximum of Brownian motion minus a parabola

TL;DR

This paper investigates the extreme tail probabilities of the maximum of Brownian motion minus a parabola, providing detailed asymptotic expansions and connecting previous related results in the literature.

Contribution

It extends existing first-order tail behavior results by deriving an asymptotic expansion for the probability that the maximum exceeds a large threshold.

Findings

Derived an asymptotic expansion for P(N>x) as x→∞.

Connected results from Groeneboom (2010) and Janson et al. (2010).

Extended the understanding of tail behavior beyond first-order approximations.

Abstract

We analyze the tail behavior of the maximum N of Brownian motion minus a parabola and give an asymptotic expansion for P(N>x) as x tends to infinity. This extends a first order result on the tail behavior, which can be deduced from Huesler and Piterbarg (1999). We also point out the relation between certain results in Groeneboom (2010) and Janson, Louchard and Martin-L\"of (2010).