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 $j^{-α}$ for some $1< α< 2 $, the trawl process exhibits long memory and its covariance decays as $j^{1-α}$. 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 $α$-stable Lévy process.