This article extends the scope of cube root asymptotics for M-estimators in two directions: allow weakly dependent observations and criterion functions drifting with the sample sizes typically due to bandwidth sequences. For dependent empirical processes that characterize criterions inducing cube root phenomena, maximal inequalities are established so that a modified continuous mapping theorem for maximizing values of the criterions delivers limit laws of the M-estimators. The limit theory is applied not only to extend existing examples, such as maximum score estimator, nonparametric maximum likelihood density estimator under monotonicity, and least median of squares, toward weakly dependent observations, but also to address some open questions, such as asymptotic properties of the minimum volume predictive region, nonparametric Hough transform estimator, and smoothed maximum score estimator for dynamic panel data.
