Limit theorems for trawl processes

Abstract

In this work we derive limit theorems for trawl processes. First, we study the asymptotic behavior of the partial sums of the discretized trawl process ${(X_{i{\mathrm{\Delta }}_n})}_{i=0}^{\lfloor nt\rfloor -1}$, under the assumption that as $n\uparrow \mathrm{\infty }$, ${\mathrm{\Delta }}_n\downarrow 0$ and $n{\mathrm{\Delta }}_n\to \mathrm{\mu }\in [0,+\mathrm{\infty }]$. Second, we prove a general result on functional convergence in distribution of trawl processes. As an application of this result, we show that a trawl process whose Lévy measure tends to infinity converges in distribution, under suitable rescaling, to a Gaussian moving average process.