On the maximum of discretely sampled fractional Brownian motion with small Hurst parameter

TL;DR

This paper investigates the distribution of the maximum of discretely sampled fractional Brownian motion with small Hurst parameter, showing it can be approximated by a normal distribution under certain conditions as the number of points grows.

Contribution

It provides a new approximation for the maximum distribution of fractional Brownian motion with small Hurst parameter over discrete sets, extending understanding of its extremal behavior.

Findings

Maximum distribution approximates a normal law with mean √ln n and variance 1/2

Approximation holds as n→∞ with slow growth, given points are not too close

Results apply to fractional Brownian motion with H→0 over discrete samples

Abstract

We show that the distribution of the maximum of the fractional Brownian motion $B^H$ with Hurst parameter $H\to 0$ over an $n$-point set $\tau \subset [0,1]$ can be approximated by the normal law with mean $\sqrt{\ln n}$ and variance $1/2$ provided that $n\to \infty$ slowly enough and the points in $\tau$ are not too close to each other.