  We consider processes with second order long range dependence resulting from
heavy tailed durations. We refer to this phenomenon as duration-driven long
range dependence (DDLRD), as opposed to the more widely studied linear long
range dependence based on fractional differencing of an $iid$ process. We
consider in detail two specific processes having DDLRD, originally presented in
Taqqu and Levy (1986), and Parke (1999). For these processes, we obtain the
limiting distribution of suitably standardized discrete Fourier transforms
(DFTs) and sample autocovariances. At low frequencies, the standardized DFTs
converge to a stable law, as do the standardized sample autocovariances at
fixed lags. Finite collections of standardized sample autocovariances at a
fixed set of lags converge to a degenerate distribution. The standardized DFTs
at high frequencies converge to a Gaussian law. Our asymptotic results are
strikingly similar for the two DDLRD processes studied. We calibrate our
asymptotic results with a simulation study which also investigates the
properties of the semiparametric log periodogram regression estimator of the
memory parameter.
