On the monotone convergence of vector means

D. R. Jensen

Journal of Multivariate Analysis, 2003, vol. 85, issue 1, 78-90

Abstract: Consider a stochastic sequence {Zn; n=1,2,...}, and define Pn([var epsilon])=P(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.

Keywords: Vector; sums; Exchangeable; vector; sequences; Majorization; Concentration; inequalities; Monotone; consistency (search for similar items in EconPapers)
Date: 2003
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