Generalized arcsine laws for fractional Brownian motion
Tridib Sadhu1, Mathieu Delorme2, Kay Jörg Wiese2
1Tata Institute of Fundamental Research, Mumbai 400005, India.
2CNRS-Laboratoire de Physique Théorique de
l'Ecole Normale Supérieure, PSL Research University, Sorbonne Universités, UPMC,
24 rue Lhomond, 75005 Paris, France.
Abstract
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 $t_{\rm last}$ the process last visits the origin, and (iii) the time $t_{\rm max}$ when it achieves its maximum (or minimum). All three observables have the same cumulative probability distribution expressed as an arcsine function, thus the name 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. $H=\tfrac{1}{2}$). We obtain the three probabilities using a perturbative expansion in $\epsilon = H-\tfrac{1}{2}$. While all three probabilities are different, this distinction can only be made at second order in $\epsilon$. Our results are confirmed to high precision by extensive numerical simulations.
arXiv:1706.01675 [pdf]
Phys. Rev. Lett. 120 (2018) 040603 [pdf]
Copyright (C)
by Kay Wiese. Last edited Feb. 3, 2018.