Conditions for the Propagation of Memory Parameter from Durations to Counts and Realized Volatility

We establish sufficient conditions on durations that arestationary with finite variance and memory parameter $d \in[0,1/2)$ to ensure that the corresponding counting process $N(t)$satisfies $Var N(t) \sim C t^{2d+1}$ ($Cgt;0$) as $t\rightarrow \infty$, with the same memory parameter $d \in[0,1/2)$ that was assumed for the durations. Thus, theseconditions ensure that the memory parameter in durationspropagates to the same memory parameter in the counts. We thenshow that any Autoregressive Conditional Duration ACD(1,1) modelwith a sufficient number of finite moments yields short memory incounts, while any Long Memory Stochastic Duration model with $dgt;0$and all finite moments yields long memory in counts, with the same$d$. Finally, we provide some results about the propagation oflong memory to the empirically-relevant case of realized varianceestimates affected by market microstructure noise contamination