The structure and analysis of spherical time-dependent processes
SIAM Journal on Applied Mathematics
Probability
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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.