# Generalized arcsine laws for fractional Brownian motion

## TL;DR

This paper extends the classical arcsine laws of Brownian motion to fractional Brownian motion, revealing how these laws are modified by non-Markovian effects using perturbative methods and numerical validation.

## Contribution

It introduces a perturbative approach to generalize the arcsine laws for fractional Brownian motion, highlighting second-order differences from standard Brownian motion.

## Key findings

- All three arcsine laws are modified for fractional Brownian motion.
- Differences between the laws appear at second order in the Hurst exponent deviation.
- Numerical simulations confirm the theoretical predictions with high precision.

## 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.