Pickands' constant at first order in an expansion around Brownian motion. (28th March 2017) - Delorme, Mathieu; Rosso, Alberto; Wiese, Kay Jörg
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 .