Statistical aspects of fuzzy monotone set-valued stochastic processes. Application to birth-and-growth processes

The paper considers a particular family of fuzzy monotone set–valued stochastic processes. The proposed setting allows us to investigate suitable α-level sets of such processes, modeling birth–and–growth processes. A decomposition theorem is established to characterize the nucleation and the growth. As a consequence, different consistent set–valued estimators are studied for growth process. Moreover, the nucleation process is studied via the hitting function, and a consistent estimator of the nucleation hitting function is derived.