On the monotone convergence of vector means

Consider a stochastic sequence {Zn; n=1,2,…}, and define Pn(ε)=P(|Zn|<ε). Then the stochastic convergence Zn→0 is said to be monotone whenever the sequence Pn(ε)↑1 monotonically in n for each ε>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 θ, then Zn=(X1+⋯+Xn)/n is not only weakly consistent for θ, 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.