Higher order representations of the Robbins–Monro process

For quasi-linear regression functions, the Robbins–Monro process Xn is decomposed in a sum of a linear form and a quadratic form both defined in the observation errors. Under regularity conditions, the remainder term is of order O(n−3/2) with respect to the Lp-norm. If a cubic form is added, the remainder term can be improved up to an order of O(n−2). As a corollary the expectation of Xn is expanded up to an error of order O(n−2). This is used to correct the bias of Xn up to an error of order O(n−3/2 log n).