Generalized Arcsine Laws for Fractional Brownian Motion

Tridib Sadhu, Mathieu Delorme, Kay Jörg Wiese
2018 Physical Review Letters  
The three arcsine laws for Brownian motion are a cornerstone of extreme-value statistics. For a Brownian B_t starting from the origin, and evolving during time T, one considers the following three observables: (i) the duration t_+ the process is positive, (ii) the time t_ last the process last visits the origin, and (iii) the time t_ max when it achieves its maximum (or minimum). All three observables have the same cumulative probability distribution expressed as an arcsine function, thus the
more » ... me of arcsine laws. We show how these laws change for fractional Brownian motion X_t, a non-Markovian Gaussian process indexed by the Hurst exponent H. It generalizes standard Brownian motion (i.e. H=12). We obtain the three probabilities using a perturbative expansion in ϵ = H-12. While all three probabilities are different, this distinction can only be made at second order in ϵ. Our results are confirmed to high precision by extensive numerical simulations.
doi:10.1103/physrevlett.120.040603 pmid:29437446 fatcat:igh6ntjjircqldselpandfcjg4