  Let $I$ be a matroidal ideal of degrre $d$ of a polynomial ring
$R=K[x_1,...,x_n]$, where $K$ is a field. Let astab$(I)$ and dstab$(I)$ be the
smallest integer $n$ for which Ass$(I^n)$ and depth$(I^n)$ stabilize,
respectively. In this paper, we show that astab$(I)=1$ if and only if
dstab$(I)=1$. Moreover, we prove that if $d=3$, then ${\rm astab}(I)={\rm
dstab}(I)$. Furthermore, we show that if $I$ is an almost square-free Veronese
type ideal of degree $d$, then ${\rm astab}(I)={\rm
dstab}(I)=\lceil\frac{n-1}{n-d}\rceil$.
