New almost sure convergence results for a special form of the multidimensional Robbins-Monro stochastic approximation procedure are developed. The results are applicable to cases where the "training data" is heavily correlated. No conditional expectation properties or boundedness assumptions are required to apply the new results. For example, when the data sequence is normal and i)M-dependent, ii) autoregressive moving average, or iii) "band-limited", the results can be used to establish the almost sure convergence of each algorithm treated. The special form of the Robbins-Monro procedure considered is motivated by a consideration of several algorithms that have been proposed for discrete-time adaptive signal-processing applications. Most of these algorithms can also be viewed as stochastic gradient-following algorithm. The ease with which the new results can be applied is illustrated.