Determining the form of the mean of a stochastic process

Abstract

Let μ be the mean function of an observable stochastic process whose sample paths fall in some Banach space with a basis and assume μ is also in this space. A procedure like Cover's ( Ann. Statist. 1, 862–871, 1973) is given which has the property that if the last nonzero coordinate of μ is the mth then with probability one this is discovered after at most a finite number of erros. If μ has an infinite number of nonzero coordinates, then with probability one this is discovered after at most a finite number of errors except for a set of μ of prior probability zero.