Pickands' constant at first order in an expansion around Brownian motion. (28th March 2017)

• Record Type: Journal Article
• Authors: Delorme, Mathieu; Rosso, Alberto; Wiese, Kay Jörg
• Is Part Of: Journal of physics. Volume 50:Number 16(2017)
• Publication Date: 2017-03-28
• Subjects: fractional Brownian motion -- Pickands constant -- extreme-value statistics; Mathematical physics -- Periodicals; Statistical physics -- Periodicals; Quantum theory -- Periodicals; Matter -- Properties -- Periodicals; 530.105
• DOI: 10.1088/1751-8121/aa5c98
• Languages: English
• ISSNs: 1751-8113

Abstract

Abstract: In the theory of extreme values of Gaussian processes, many results are expressed in terms of the Pickands constant H α . This constant depends on the local self-similarity exponent α of the process, i.e. locally it is a fractional Brownian motion (fBm) of Hurst index H = α / 2 . Despite its importance, only two values of the Pickands constant are known: H 1 = 1 and H 2 = 1 / π . Here, we extend the recent perturbative approach to fBm to include drift terms. This allows us to investigate the Pickands constant H α around standard Brownian motion ( α = 1 ) and to derive the new exact result H α = 1 − ( α − 1 ) γ E + O ( α − 1 ) 2 .