Universal nonparametric kernel-type estimators for the mean and covariance functions of a stochastic process

Let $f_1(t), \dots, f_n(t)$ be independent copies of some a.s. continuous stochastic process $f(t)$, $t\in[0,1]$, which are observed with noise. We consider the problem of nonparametric estimation of the mean function $\mu(t) = \mathbf{E}f(t)$ and of the covariance function $\psi(t,s)=\operatorname{Cov}\{f(t),f(s)\}$ if the noise values of each of the copies $f_i(t)$, $i=1,\dots,n$, are observed in some collection of generally random (in general) time points (regressors). Under wide assumptions on the time points, we construct uniformly consistent kernel estimators for the mean and covariance functions both in the case of sparse data (where the number of observations for each copy of the stochastic process is uniformly bounded) and in the case of dense data (where the number of observations at each of $n$ series is increasing as $n\to\infty$). In contrast to the previous studies, our kernel estimators are universal with respect to the structure of time points, which can be either fixed rather than necessarily regular, or random rather than necessarily formed of independent or weakly dependent random variables.