Estimating functionals of a stochastic process

The problem of estimating the integral of a stochastic process from observations at a finite number N of sampling points has been considered by various authors. Recently, Benhenni and Cambanis (1992) studied this problem for processes with mean 0 and Holder index K + 1/2, K is an element of N. These results are here extended to processes with arbitrary Holder index. The estimators built here are linear in the observations and do not require the a priori knowledge of the smoothness of the process. If the process satisfies a Holder condition with index s in quadratic mean, we prove that the rate of convergence of the mean square error is N2s+1 as N goes to infinity, and build estimators that achieve this rate.