Passive stochastic approximation

It is required to find the root of the equation $f(x)=0$ when the values of $f(x)$ are measured with a random error in random points whose choice cannot be controlled; The recurrent Hardle—Nixdorf method for solution of this problem is investigated. Its convergence almost surely and in the mean square sense are proved, the convergence rate is estimated. A technique is proposed for choice of optimal parameters of the method which is proved to lead to the lowerbound (in terms of the order of magnitude) of the accuracy of arbitrary methods for solution of the problem.