Selfdecomposability and selfsimilarity: a concise primer

TL;DR

This paper provides a concise overview of the relationships among infinitely divisible, selfdecomposable, and stable laws, emphasizing their roles in Levy processes, stationarity, selfsimilarity, and Ornstein-Uhlenbeck processes driven by Levy noise.

Contribution

It offers a clear, summarized connection between these classes of laws and their applications in stochastic processes, highlighting new insights into selfdecomposability and stationarity.

Findings

Selfdecomposable laws are key in Levy process modeling.

Ornstein-Uhlenbeck processes with Levy noise have selfdecomposable stationary distributions.

The paper clarifies the relationships among infinitely divisible, selfdecomposable, and stable laws.

Abstract

We summarize the relations among three classes of laws: infinitely divisible, selfdecomposable and stable. First we look at them as the solutions of the Central Limit Problem; then their role is scrutinized in relation to the Levy and the additive processes with an emphasis on stationarity and selfsimilarity. Finally we analyze the Ornstein-Uhlenbeck processes driven by Levy noises and their selfdecomposable stationary distributions, and we end with a few particular examples.