Asymptotic behaviour of the distribution density of the fractional L\'evy motion

TL;DR

This paper studies the asymptotic behavior of the distribution density of fractional Lévy motion, revealing distinct characteristics in short and long memory regimes based on the Hurst parameter.

Contribution

It provides a detailed analysis of the asymptotic distribution density behavior of fractional Lévy motion for different Hurst parameter ranges, highlighting their differences.

Findings

Density behavior differs significantly between short and long memory cases.

Asymptotic formulas are derived for the distribution density.

Examples illustrate the contrasting behaviors in different Hurst regimes.

Abstract

We investigate the distribution properties of the fractional L\'evy motion. We consider separately the cases $0<H<1/2$ (short memory) and $1/2<H<1$ (long memory), where $H$ is the Hurst parameter, and present the asymptotic behaviour of the distribution density of the process. Some examples are provided, in which it is shown that the behaviour of the density in the cases $0<H<1/2$ and $1/2<H<1$ is completely different.