A Fubini-type limit theorem for the integrated hyperuniform infinitely divisible moving averages

Abstract

This short note shows a limiting behavior of integrals of some centered antipersistent stationary infinitely divisible moving averages as the compact integration domain in $d\ge 1$ dimensions extends to the whole positive quadrant $\mathbb{R}^d_+$. Namely, the weak limit of their finite dimensional distributions is again a moving average with the same infinitely divisible purely jump integrator measure (i.e., possessing no Gaussian component), but with an integrated kernel function. The results apply equally to time series ($d=1$) as well as to random fields ($d>1$). Apart from the existence of the expectation, no moment assumptions on the moving average are imposed allowing it to have an infinite variance as e.g. in the case of $\alpha$-stable moving averages with $\alpha\in(1,2)$ . If the field is additionally square integrable, its covariance integrates to zero (hyperuniformity).