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The tail of the maximum of Brownian motion minus a parabola. (English) Zbl 1244.60052

Summary: We analyze the tail behavior of the maximum \(N\) of \(\{W(t)-t^2:t\geq0\}\), where \(W\) is standard Brownian motion on \([0,\infty)\), and give an asymptotic expansion for \(\text{P}\{N\geq x\}\), as \(x\to\infty\). This extends a first order result on the tail behavior, which can be deduced from results by J. Hüsler and V. Piterbarg [Stochastic Processes Appl. 83, No. 2, 257–271 (1999; Zbl 0997.60057)]. We also point out the relation between certain results in [P. Groeneboom, Electron. J. Probab. 15, 1930–1937 (2010; Zbl 1226.60110); S. Janson, G. Louchard and A. Martin-Löf, ibid. 15, 1893–1929 (2010; Zbl 1226.60111)].

MSC:

60G70 Extreme value theory; extremal stochastic processes
60J65 Brownian motion

Software:

Mathematica; DLMF
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