  In this paper, sufficient conditions are given for the existence of limiting
distribution of a conservative affine process on the canonical state space
$\mathbb{R}_{\geqslant0}^{m}\times\mathbb{R}^{n}$, where $m,\thinspace
n\in\mathbb{Z}_{\geqslant0}$ with $m+n>0$. Our main theorem extends and unifies
some known results for OU-type processes on $\mathbb{R}^{n}$ and
one-dimensional CBI processes (with state space $\mathbb{R}_{\geqslant0}$). To
prove our result, we combine analytical and probabilistic techniques; in
particular, the stability theory for ODEs plays an important role.
