Scaling limits of stationary determinantal shot-noise fields

We consider a shot-noise field defined on a stationary determinantal point process on $\mathbb{R}^d$ associated with i.i.d. amplitudes and a bounded response function, for which we investigate the scaling limits as the intensity of the point process goes to infinity. Specifically, we show that the centralized and suitably scaled shot-noise field converges in finite dimensional distributions to i) a Gaussian random field when the amplitudes have the finite second moment and ii) an $\alpha$-stable random field when the amplitudes follow a regularly varying distribution with index $-\alpha$ for $\alpha\in(1,2)$. We first prove the corresponding results for the shot-noise field defined on a homogeneous Poisson point process and then extend them to the one defined on a stationary determinantal point process.