  We establish sufficient conditions on durations that are stationary with
finite variance and memory parameter $d \in [0,1/2)$ to ensure that the
corresponding counting process $N(t)$ satisfies $\textmd{Var} N(t) \sim C
t^{2d+1}$ ($C>0$) as $t \to \infty$, with the same memory parameter $d \in
[0,1/2)$ that was assumed for the durations. Thus, these conditions ensure that
the memory in durations propagates to the same memory parameter in counts and
therefore in realized volatility. We then show that any utoregressive
Conditional Duration ACD(1,1) model with a sufficient number of finite moments
yields short memory in counts, while any Long Memory Stochastic Duration model
with $d>0$ and all finite moments yields long memory in counts, with the same
$d$.
