Sojourn functionals of time-dependent $\chi^2$-random fields on two-point homogeneous spaces

We investigate geometric properties of invariant spatio-temporal random fields X : Md × R → R defined on a compact two-point homogeneous space Md in any dimension d ≥ 2, and evolving over time. In particular, we focus on chi-squared-distributed random fields, and study the large-time behavior (as T → +∞) of the average on [0,T] of the volume of the excursion set on the manifold, i.e. of {X(·, t) ≥ u} (for any u > 0). The Fourier components of X may have short or long memory in time, i.e. integrable or non-integrable temporal covariance functions. Our argument follows the approach developed in Marinucci et al. (2021) and allows us to extend their results for invariant spatio-temporal Gaussian fields on the two-dimensional unit sphere to the case of chisquared distributed fields on two-point homogeneous spaces in any dimension. We find that both the asymptotic variance and limiting distribution, as T → +∞, of the average empirical volume turn out to be non-universal, depending on the memory parameters of the field X.