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Passive stochastic approximation. (English. Russian original) Zbl 0719.62092
Autom. Remote Control 50, No. 11, 1563-1569 (1989); translation from Avtom. Telemekh. 1989, No. 11, 127-134 (1989).
Summary: A root of the equation $$f(x)=0$$ is sought in the case that the values of f(x) are measured with a random error at random points whose choice cannot be controlled. The recursive Härdle-Nixdorf method [W. K. Härdle and R. Nixdorf, IEEE Trans. Inf. Theory IT-33, 367-372 (1987; Zbl 0636.62079)] of solution of this problem is studied. Its almost surely convergence and in the mean square is proved, and the convergence rate is estimated. A technique of selection of the optimal parameters of this method is described, and it is proved that this yields a lower bound (in order of magnitude) for the accuracy of any method of solution of the problem.

##### MSC:
 62L20 Stochastic approximation 65C99 Probabilistic methods, stochastic differential equations