Asymptotical properties of distributions of isotropic L\' evy processes

TL;DR

This paper derives precise asymptotic behaviors for the tail probabilities and transition densities of isotropic Lévy processes and subordinators, expressed in terms of their characteristic exponents and derivatives, covering various scaling orders.

Contribution

It provides new asymptotic formulas for isotropic Lévy processes and subordinators, extending understanding of their tail behaviors in terms of characteristic exponents.

Findings

Asymptotic expressions for tail probabilities and densities are established.

Results apply to processes with scaling order between 0 and 2, including 2.

Explicit formulas relate tail behavior to the radial part of the characteristic exponent.

Abstract

In this paper, we establish the precise asymptotic behaviors of the tail probability and the transition density of a large class of isotropic L\'evy processes when the scaling order is between 0 and 2 including 2. We also obtain the precise asymptotic behaviors of the tail probability of subordinators when the scaling order is between 0 and 1 including 1. The asymptotic expressions are given in terms of the radial part of characteristic exponent $\psi$ and its derivative. In particular, when $\psi(\lambda)-\frac{\lambda}{2}\psi'(\lambda)$ varies regularly, as $\frac{t\psi(r^{-1})^2}{\psi(r^{-1})-(2r)^{-1}\psi'(r^{-1})} \to 0$ the tail probability $\mathbb{P}(|X_t|\geq r)$ is asymptotically equal to a constant times $ t( \psi(r^{-1})-(2r)^{-1}\psi'(r^{-1})).$