  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.
