Convergence of Stochastic-Approximation Procedures in the Case of a Regression Equation with Several Roots

It is shown that the Robbins–Monro stochastic-approximation procedure cannot converge to the root $\tilde{x}$ of the regression equation $R(x)=0$ if at least one eigenvalue of the matrix $\frac{\partial R}{\partial x}(\tilde{x})$ has a positive real part. A similar result is obtained for the Kiefer–Wolfowitz procedure.