Discrete-time trawl processes

Abstract

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 et al. (2011, 2014).In the case when the trawl function decays as a power function of the lag with exponent 1<α<2, the trawl process exhibits long memory and its covariance function is non-summable. 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. Moreover if the trawl function admits a faster decay rate, then the classical Donsker’s invariance principle holds true.