We study the convergence of a class of stable online algorithms for stochastic convex optimization in settings where we do not receive independent samples from the distribution over which we optimize, but instead receive samples that are coupled over time. We show the optimization error of the averaged predictor output by any stable online learning algorithm is upper bounded|with high probability|by the average regret of the algorithm, so long as the underlying stochastic process is - or -mixing. We additionally show sharper convergence rates when the expected loss is strongly convex, which includes as special cases linear prediction problems including linear and logistic regression, least-squares SVM, and boosting.