A central limit theorem for the volume of the excursion sets of associated stochastically continuous stationary random fields

In [1], the multivariate central limit theorem (CLT) for the volumes of excursion sets of stationary quasi-associated random fields on $\mathbb R^d$ is proved. We extend their results by considering associated stochastically continuous stationary random fields which need not necessarily have second moments. Based on the CLT, we construct a statistical hypothesis test and provide simple conditions on the tail behavior of the kernel functions for certain max-stable and sum-stable random fields with spectral representation such that the assumptions of the CLT are satisfied.
[1] Bulinski, A., Spodarev, E. and Timmermann, F., Central limit theorems for the excursion sets volumes of weakly dependent random fields, accepted at Bernoulli, 2010.