Wear convergence of stochastic approximation processes with random indices

Conditions are given for weak convergence through random indices of a general stochastic approximation process which includes the Robbins-Monro and Kiefer-Wolfowitz processes. For a particular index, a sequential fixed-width bounded length confidence interval for the parameter being estimated is established. As an example, an optimal recursive estimator and confidence interval for the mode of a distribution function is constructed.