Data depth for measurable noisy random functions

In the literature on data depth applicable to random functions, it is usually assumed that the trajectories of all the random curves are continuous, known at each point of the domain, and observed exactly. These assumptions turn out to be unrealistic in practice, as the functions are often observed only on a finite grid of time points, and in the presence of measurement errors. In this work, we provide the necessary theoretical background enabling the extension of the statistical methodology based on data depth to measurable (not necessarily continuous) random functions observed within the latter framework. It is shown that even if the random functions are discontinuous, observed discretely, and contaminated with additive noise, many common depth functionals maintain the fine consistency properties valid in the ideal case of completely observed noiseless functions. For the integrated depth for functions, we provide uniform rates of convergence over the space of integrable functions.