Sojourn functionals of time-dependent χ²-random fields on two-point homogeneous spaces

2024/03/26 by Caponera, Alessia, Rossi, Maurizia, Medina, María Dolores Ruiz · 1 citation
#FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.2403.17538

Abstract

In this note we investigate geometric properties of invariant spatio-temporal random fields X:\mathbb M^d× \mathbb R→ \mathbb R defined on a compact two-point homogeneous space \mathbb M^d 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 \lbrace X(⋅, t)≥ u\rbrace (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, Rossi, Vidotto (2021) Ann. Appl. Probab.) and allow to extend their results for invariant spatio-temporal Gaussian fields on the two-dimensional unit sphere to the case of chi-squared 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.

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