Sharp convergence rates of empirical unbalanced optimal transport for spatio-temporal point processes

We statistically analyze empirical plug-in estimators for unbalanced optimal transport (UOT) formalisms, focusing on the Kantorovich–Rubinstein distance, between general intensity measures based on observations from spatio-temporal point processes. Specifically, we model the observations by two weakly time-stationary point processes with spatial intensity measures μ and ν over the expanding window (0, t] as t increases to infinity, and establish sharp convergence rates of the empirical UOT in terms of the intrinsic dimension of the measures. We assume a sub-quadratic temporal growth condition of the variance of the process, which allows for a wide range of temporal dependencies. As the growth approaches quadratic, the convergence rate becomes slower. This variance assumption is related to the time-reduced factorial covariance measure, and we exemplify its validity for various point processes, including the Poisson cluster, Hawkes, Neyman–Scott, and log-Gaussian Cox processes. Complementary to our upper bounds, we also derive matching lower bounds for various spatio-temporal point processes of interest and establish near minimax rate optimality of the empirical Kantorovich–Rubinstein distance.