  We introduce a class of discrete time stationary trawl processes taking real
or integer values and written as sums of past values of independent `seed'
processes on shrinking intervals (`trawl heights'). Related trawl processes in
continuous time were studied in Barndorff-Nielsen (2011) and Barndorff-Nielsen
et al. (2014), however in our case, the i.i.d. seed processes can be very
general and need not be infinitely divisible. In the case when the trawl height
decays with the lag as $j^{-\alpha}$ for some $1< \alpha < 2 $, the trawl
process exhibits long memory and its covariance decays as $j^{1-\alpha}$. We
show that under general conditions on generic seed process, the normalized
partial sums of such trawl process may tend either to a fractional Brownian
motion or to an $\alpha$-stable L\'evy process.
