  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).
