Stable Central Limit Theorems for Super Ornstein-Uhlenbeck Processes, II

TL;DR

This paper extends previous work on super Ornstein-Uhlenbeck processes by proving stable central limit theorems for all polynomial growth functions, revealing independence of limit variables across regimes.

Contribution

It establishes stable CLTs for all polynomial growth functions in super Ornstein-Uhlenbeck processes, generalizing prior results and demonstrating independence of limit variables.

Findings

Stable CLTs hold for all polynomial growth functions.

Limit variables in different regimes are independent.

The results unify and extend previous partial CLTs.

Abstract

This paper is a continuation of our recent paper (Elect. J. Probab. 24 (2019), no. 141) and is devoted to the asymptotic behavior of a class of supercritical super Ornstein-Uhlenbeck processes $(X_t)_{t\geq 0}$ with branching mechanisms of infinite second moment. In the aforementioned paper, we proved stable central limit theorems for $X_t(f) $ for some functions $f$ of polynomial growth in three different regimes. However, we were not able to prove central limit theorems for $X_t(f) $ for all functions $f$ of polynomial growth. In this note, we show that the limit stable random variables in the three different regimes are independent, and as a consequence, we get stable central limit theorems for $X_t(f) $ for all functions $f$ of polynomial growth.