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

TL;DR

This paper establishes a Fubini-type limit theorem for integrated hyperuniform infinitely divisible moving averages, describing their asymptotic behavior as the domain expands to infinity, applicable to both time series and random fields.

Contribution

It introduces a new limit theorem for hyperuniform infinitely divisible moving averages, extending understanding of their asymptotic behavior without moment restrictions.

Findings

Weak limit of finite-dimensional distributions is a moving average with the same jump measure.

Results apply to both time series and random fields.

The theorem holds even for infinite variance cases like stable processes.

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).