  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.
