Discrete time trawl processes constitute a large class of time series parameterized by a trawl sequence (a j) j$in$N and defined though a sequence of independent and identically distributed (i.i.d.) copies of a continuous time process ($gamma$(t)) t$in$R called the seed process. They provide a general framework for modeling linear or non-linear long range dependent time series. We investigate the spectral estimation, either pointwise or broadband, of long range dependent discrete-time trawl processes. The difficulty arising from the variety of seed processes and of trawl sequences is twofold. First, the spectral density may take different forms, often including smooth additive correction terms. Second, trawl processes with similar spectral densities may exhibit very different statistical behaviors. We prove the consistency of our estimators under very general conditions and we show that a wide class of trawl processes satisfy them. This is done in particular by introducing a weighted weak dependence index that can be of independent interest. The broadband spectral estimator includes an estimator of the long memory parameter. We complete this work with numerical experiments to evaluate the finite sample size performance of this estimator for various integer valued discrete time trawl processes.
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