Long memory was first defined as the fact that stationary processes have non summable covariances. In 1961 Murray Rosenblatt exhibited the long memory properties of functions of a Gaussian process. Namely the partial sums process renormalised with n^H for for H>1/2 may converge to some non Gaussian process. The aim of the talk is to consider a new class of such models with possibly integer values.
A special case of trawl processes is easily proved to have long memory. Various estimation techniques are first discussed. Then one may prove that both a fBm limit or a stable Lévy limit can be obtained as limits of the partial sums process in the current case. The latter Lévy limit is obtained for integer valued models. Other issues are also on progress.