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On the monotone convergence of vector means

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  • Jensen, D. R.

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.

Suggested Citation

  • Jensen, D. R., 2003. "On the monotone convergence of vector means," Journal of Multivariate Analysis, Elsevier, vol. 85(1), pages 78-90, April.
    • Handle: RePEc:eee:jmvana:v:85:y:2003:i:1:p:78-90
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    References listed on IDEAS

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    1. Cambanis, Stamatis & Huang, Steel & Simons, Gordon, 1981. "On the theory of elliptically contoured distributions," Journal of Multivariate Analysis, Elsevier, vol. 11(3), pages 368-385, September.
    2. Jensen, D. R., 1997. "Peakedness of linear forms in ensembles and mixtures," Statistics & Probability Letters, Elsevier, vol. 35(3), pages 277-282, October.
    3. D. Jensen, 1996. "Straight-line models in star-shaped mixtures," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 44(1), pages 101-117, December.
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