Optimal asymptotic quadratic error of nonparametric regression function estimates for a continuous-time process from sampled-data

For different classes of deterministic and random sampling (t(k)), we estab lish the asymptotic expressions for the bias and the variance of the estima te r(n)(x) based on sampled data (X-tk, Y-tk)k=1,...,n for the regression f unction r(x) = E(Y-t/X-t = x) of unbounded continuous-time processes (X-t, Y-t)(t is an element of R) (not necessarily stationary). Under mild mixing conditions, we show that r(n)(x) has exactly the same asymptotic quadratic error as in the i.i.d. case. In order to prove this result, we use some lar ge deviations inequalities for mixing processes.