Consider a stochastic sequence {Zn; n=1,2,...}, and define Pn([var epsilon])=P(Zn<[var epsilon]). Then the stochastic convergence Zn-->0 is said to be monotone whenever the sequence Pn([var epsilon])[short up arrow]1 monotonically in n for each [var epsilon]>0. This mode of convergence is investigated here; it is seen to be stronger than convergence in quadratic mean; and scalar and vector sequences exhibiting monotone convergence are demonstrated. In particular, if {X1,...,Xn} is a spherical Cauchy vector whose elements are centered at [theta], then Zn=(X1+...+Xn)/n is not only weakly consistent for [theta], but it is shown to follow a monotone law of large numbers. Corresponding results are shown for certain ensembles and mixtures of dependent scalar and vector sequences having n-extendible joint distributions. Supporting facts utilize ordering by majorization; these extend several results from the literature and thus are of independent interest.
