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Non-stationary self-similar Gaussian processes as scaling limits of power law shot noise processes and generalizations of fractional Brownian motion
We study shot noise processes with Poisson arrivals and non-stationary noises. The noises are conditionally independent given the arrival times, but the distribution of each noise does depend on its arrival time. We establish scaling limits for such shot noise processes in two situations: 1) the conditional variance functions of the noises have a power law and 2) the conditional noise distributions are piecewise. In both cases, the limit processes are self-similar Gaussian with nonstationary increments. Motivated by these processes, we introduce new classes of self-similar Gaussian processes with non-stationary increments, via the time-domain integral representation, which are natural generalizations of fractional Brownian motions.Published versio
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 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
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