Asymptotics for duration-driven long range dependent processes

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 i.i.d. process. We consider in detail two specific processes having DDLRD, originally presented in Taqqu and Levy [1986. Using renewal processes to generate long-range dependence and high variability. Dependence in Probability and Statistics. Birkhauser, Boston, pp. 73-89], and Parke [1999. What is fractional integration? Review of Economics and Statistics 81, 632-638]. 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.