Levy's distributional property for symmetric Levy processes

TL;DR

This paper extends Levy's distributional property to symmetric Levy processes, broadening the classical theorem from Brownian motion to more general stochastic processes with jumps.

Contribution

It generalizes Levy's distributional property from Brownian motion to a wider class of symmetric Levy processes with specific generating triplets.

Findings

Levy's property holds for symmetric Levy processes with specified triplets.

The generalization encompasses processes with jumps, not just continuous paths.

Provides a theoretical foundation for analyzing symmetric Levy processes.

Abstract

We present the Levy's distributional property for symmetric Levy processes with generating triplet $(0, 0,\nu)$ or $(\sigma>0, \gamma, \nu)$ where $\nu$ is a symmetric measure on $R\backslash\{0\}$. This generalizes the classical Levy's theorem about Brownian motions with drift.