Excursions of random fields with long range dependence
Final Report Abstract
We investigate short memory conditions and the asymptotic behaviour of iintegrals of infintely divisible moving average random fields with (possibly) infinite variance within growing observation windows. First, we prove a sufficient condition for the short range dependence of measurable stationary infinitely divisible moving average random fields with d–dimensional index space. Here, the short/long range dependence concept based on the covariance of excursions is borrowed from the paper of Kulik & Spodarev (2021). In the special case of symmetric stable moving averages, our new condition coincides with the one from paper Makogin et al. (2021). Second, we show a limiting behavior of integrals of some centered antipersistent stationary infinitely divisible moving averages as the compact integration domain in d ≥ 1 dimensions extends to the whole positive quadrant 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 α–stable moving averages with α ∈ (1, 2) . If the field is additionally square integrable, its covariance integrates to zero (hyperuniformity).
