ESTIMATING FUNCTIONALS OF A STOCHASTIC-PROCESS

The problem of estimating the integral of a stochastic process from ob servations at a finite number N of sampling points has been considered by various authors. Recently, Benhenni and Cambanis (1992) studied th is 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 arbit rary Holder index. The estimators built here are linear in the observa tions and do not require the a priori knowledge of the smoothness of t he process. If the process satisfies a Holder condition with index s i n quadratic mean, we prove that the rate of convergence of the mean sq uare error is N2s+1 as N goes to infinity, and build estimators that a chieve this rate.