Convergence rate of stochastic approximation algorithms in the degenerate case

Let f(.) be an unknown function whose root x(0) is sought by stochastic approximation (SA). Convergence rate and asymptotic normality are usually established for the nondegenerate case f'(x(0)) not equal 0. This paper demonstrates the convergence rate of SA algorithms for the degenerate case f'(x(0)) = 0. In comparison with the previous work, in this paper no growth rate restriction is imposed on f(.), no statistical property is required for the measurement noise, the general step size is considered, and the result is obtained for the multidimensional case, which is not a straightforward extension of the one-dimensional result. Although the observation noise may be either deterministic or random, the analysis is purely deterministic and elementary.