Convergence of Continuous and Discrete Robbins-Monro Procedures in the Case of a Multiple-Root Regression Equation

The limiting behavior of a continuous- or discrete-time stochastic approximation process is investigated in the case of a regression equation having several roots. Subject to certain assumptions, a hypothesis advanced in [V. Fabian, Czech. Math. J., 1960, vol. 10, no. 2, pp. 123–159; Trans. 3rd Prague Conf. on Information Theory, Statistical Decision Functions, and Random Processes, Prague, 1964, pp. 85–105] is proved, namely that unstable points of a deterministic system corresponding to a stochastic approximation process cannot be with positive probability limit points for that process.