Mean function estimation for a noisy random process under a sparse data condition

We consider a regression statement of the problem of estimating the mean function of some almost sure continuous random process, when noisy values of independent copies of this random process are observed in some known sets of time points (generally speaking, random). Moreover, the size of observations for each of the copies is random, and the total collection of the time points for all series does not necessarily consist of independent and identically distributed random variables. This setting includes two of the most popular sparse data variants in the scientific literature, in which ever the sizes of observations in the series are independent identically distributed random variables, or the sizes of observations in each series are nonrandom and uniformly bounded over all series.
The paper proposes new kernel-type estimators for the mean function of a random process. The uniform consistency of the new kernel estimators is proved under very weak and universal restrictions regarding the stochastic nature of observed time points: it is only required that the entire set of these points with a high probability would form a refining partition of the original random process domain.