On the monotone convergence of vector means

✍ Scribed by D.R. Jensen

• Book ID: 104269842
• Publisher: Elsevier Science
• Year: 2003
• Tongue: English
• Weight: 209 KB
• Volume: 85
• Category: Article
• ISSN: 0047-259X
• DOI: 10.1016/s0047-259x(02)00039-8

✦ Synopsis

Consider a stochastic sequence fZ n ; n ¼ 1; 2; yg; and define P n ðeÞ ¼ PðjZ n joeÞ: Then the stochastic convergence Z n -0 is said to be monotone whenever the sequence P n ðeÞm1 monotonically in n for each e40: 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 fX 1 ; y; X n g is a spherical Cauchy vector whose elements are centered at y; then Z n ¼ ðX 1 þ ? þ X n Þ=n is not only weakly consistent for y; 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.