Stochastic approximation (SA) is a general framework for analyzing the convergence of a large collection of stochastic root-finding algorithms. The Kiefer-Wolfowitz and stochastic gradient algorithms are two well-known (and widely used) examples of SA. Because of their applicability to a wide range of problems, many results have been obtained regarding the convergence properties of SA procedures. One important reference in the literature, Fabian (1968), derives general conditions for the asymptotic normality of the SA iterates. Since then, many results regarding asymptotic normality of SA procedures have relied heavily on Fabian's theorem. Unfortunately, some of the assumptions of Fabian's result are not applicable to some modern implementations of SA in control and learning. In this paper we explain the nature of this incompatibility and show how Fabian's theorem can be generalized to address the issue. (C) 2018 Elsevier Ltd. All rights reserved.