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Discrete time trawl processes constitute a large class of time seriesparameterized by a trawl sequence (a j) j$\in$N and defined though a sequenceof independent and identically distributed (i.i.d.) copies of a continuous timeprocess ($\gamma$(t)) t$\in$R called the seed process. They provide a generalframework for modeling linear or non-linear long range dependent time series.We investigate the spectral estimation, either pointwise or broadband, of longrange dependent discrete-time trawl processes. The difficulty arising from thevariety of seed processes and of trawl sequences is twofold. First, thespectral density may take different forms, often including smooth additivecorrection terms. Second, trawl processes with similar spectral densities mayexhibit very different statistical behaviors. We prove the consistency of ourestimators under very general conditions and we show that a wide class of trawlprocesses satisfy them. This is done in particular by introducing a weightedweak dependence index that can be of independent interest. The broadbandspectral estimator includes an estimator of the long memory parameter. Wecomplete this work with numerical experiments to evaluate the finite samplesize performance of this estimator for various integer valued discrete timetrawl processes.

Spectral estimation for non-linear long range dependent discrete time trawl processes