A note on small probabilities for spherical random fields at a critical regime

TL;DR

This paper investigates the behavior of small probabilities in spherical Gaussian random fields, establishing Chung's law of the iterated logarithm and analyzing how the decay rate depends on spectral properties.

Contribution

It provides new results on small probability decay rates for spherical Gaussian fields, linking them to the high-frequency behavior of the angular power spectrum.

Findings

Chung's law of the iterated logarithm is established for these fields.

The decay speed of small ball probabilities increases with decreasing memory or space parameters.

Results depend on the high-frequency behavior of the angular power spectrum.

Abstract

We consider time-dependent space isotropic and time stationary spherical Gaussian random fields. We establish Chung's law of the iterated logarithm and solve the small probabilities problem. Our results depend on the high-frequency behaviour of the angular power spectrum: the speed of decay of the small ball probability is faster as either the memory parameter or the space-parameter decreases.