Relevant sampling of the short-time Fourier transform of time-frequency localized functions

We study the random sampling of the short-time Fourier transform of functions that are localized in a compact region in the time-frequency plane. We follow the approach introduced by Bass and Gr"ochenig for band-limited functions, and show that with a high, controllable probability, a sufficiently dense set of local random samples from the region of concentration in the time-frequency plane yields a sampling inequality for the short-time Fourier transform of time-frequency localized functions on the region.