We establish sufficient conditions on durations that
are stationary with finite variance and memory
parameter to ensure that
the corresponding counting process
N(t) satisfies
Var N(t) ~
Ct2d+1
(C > 0) as t
→ ∞, with the same memory parameter that was assumed
for the durations. Thus, these conditions ensure
that the memory parameter in durations propagates to
the same memory parameter in the counts. We then
show that any autoregressive conditional duration
ACD(1,1) model with a sufficient number of finite
moments yields short memory in counts, whereas any
long memory stochastic duration model with
d > 0 and all finite moments
yields long memory in counts, with the same
d. Finally, we provide some
results about the propagation of long memory to the
empirically relevant case of realized variance
estimates affected by market microstructure noise
contamination.