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Limit theorems for trawl processes. (English) Zbl 1477.60057

The topic of the article are the probabilistic limit theorems for trawl processes, stationary infinitely divisible stochastic processes introduced in [O. E. Barndorff-Nielsen, Braz. J. Probab. Stat. 25, No. 3, 294–322 (2011; Zbl 1244.60037)] (see also [O. E. Barndorff-Nielsen et al., Scand. J. Stat. 41, No. 3, 693–724 (2014; Zbl 1309.60033)]).
The first result concerns the asymptotic behavior of the partial sums \[ \mathbf{S}^n=\left(\sum\limits_{k=0}^{\lfloor nt\rfloor-1}(X_{\triangle_nk}-\mathbf{E}(X_{\triangle_nk}))\right)_{t>0}, \] where \({\triangle_n\downarrow0}\) and \({n\triangle_n\to\mu}, {\mu\in[0,+\infty]}\) as \({n\uparrow\infty}\). Three different cases are considered. If \({\mu\in(0,+\infty)}\) then it relates with a Riemann sum and converges in probability to \({\int_0^{t\mu}(X_s-\mathbf{E}(X_s))\,ds}.\) For \({\mu=0}\) the behavior of \(\mathbf{S}^n\) depends on the increments of \(X\) around 0 and there is proved that after properly rescaling it converges stably to some stochastic integral driven by a Lévy process. When \({\mu=+\infty}\) the limit depends on whether the trawl process \(X\) has short or long memory. Consequently, \(\mathbf{S}^n\) typically converges either to a Brownian motion or to a fractional Brownian motion.
The second main result is a general functional limit theorem for trawl processes in terms of the characteristic triplets of their Levy seeds.

MSC:

60F17 Functional limit theorems; invariance principles
60G10 Stationary stochastic processes
60G57 Random measures
60J65 Brownian motion
60G22 Fractional processes, including fractional Brownian motion
60G51 Processes with independent increments; Lévy processes
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