Kernel estimators for the mean function of a stochastic process under sparse design conditions

The problem of nonparametric estimation of the mean function for a continuous random process is considered, when the noisy values of each of its independent trajectories are observed in some random time points — design elements. Under broad conditions on the dependence of design elements, uniformly consistent estimates are constructed for the mean function in the case of one of the versions of so-called sparse design, when the number of design elements for each of the trajectories is the same and independent of the growing number of series of observations. Unlike the papers of predecessors, we do not require that the set of design elements consist of independent identically distributed or weakly dependent random variables. Regarding the design, it is only assumed that the entire set of design points with a high probability forms a refining partition of the domain of the random process under consideration.