Non-universal fluctuations of the empirical measure for isotropic stationary fields on S2×R

Abstract

In this paper, we consider isotropic and stationary real Gaussian random fields defined on ${\mathbb{S}}^2\times \mathbb{R}$ and we investigate the asymptotic behavior, as $\mathit{T}\to +\infty$, of the empirical measure (excursion area) in ${\mathbb{S}}^2\times [0,\mathit{T}]$ at any threshold, covering both cases when the field exhibits short and long memory, that is, integrable and nonintegrable temporal covariance. It turns out that the limiting distribution is not universal, depending both on the memory parameters and the threshold. In particular, in the long memory case a form of Berry’s cancellation phenomenon occurs at zero-level, inducing phase transitions for both variance rates and limiting laws.