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

This paper is a continuation of our recent paper(Electron. J. Probab., 24(141),(2019)) and is devoted to the asymptotic behavior of a class of supercritical super Ornstein–Uhlenbeck processes(Xt)t≥0 with branching mechanisms of infinite second moments. In the aforementioned paper, we proved stable central limit theorems for Xt(f) for some functions f of polynomial growth in three different regimes. However, we were not able to prove central limit theorems for Xt(f) for all functions f of polynomial growth. In this note, we show that the limiting stable random variables in the three different regimes are independent, and as a consequence, we get stable central limit theorems for Xt(f)for all functions f of polynomial growth.