Abstract
For arrays of rowwise pairwise negative quadrant dependent random variables, conditions are provided under which weighted averages converge in mean to 0 thereby extending a result of Chandra, and conditions are also provided under which normed and centered row sums converge in mean to 0. These results are new even if the random variables in each row of the array are independent. Examples are provided showing (i) that the results can fail if the rowwise pairwise negative quadrant dependent hypotheses are dispensed with, and (ii) that almost sure convergence does not necessarily hold.
Highlights
For a sequence of independent and identically distributed (i.i.d.) random variables {Xn, n ≥ 1} with EX1 = 0, Pyke and Root [12] established the degenerate mean convergence law n j=1 Xj −L→1 0. n (1.1)A considerably simpler proof of the limit law (1.1) was obtained by Dharmadhikari [4] who did not refer to the Pyke and Root [12] article
We extend in Theorems 3.1 and 3.2 this degenerate mean convergence theorem of Chandra [3] in two directions: (i) Our results pertain to weighted averages either from an array of random variables whose nth row is comprised of kn pairwise negative quadrant dependent random variables, n ≥ 1 (Theorem 3.1) or from an array of random variables whose nth row is comprised of kn pairwise independent random variables, n ≥ 1 (Theorem 3.2)
Theorem 3.1 Let {Xn,j, 1 ≤ j ≤ kn, n ≥ 1} be an array of rowwise pairwise negative quadrant dependent (PNQD) mean 0 random variables which is stochastically dominated by a random variable X with E|X| < ∞
Summary
Theorem 3.1 Let {Xn,j, 1 ≤ j ≤ kn, n ≥ 1} be an array of rowwise PNQD mean 0 random variables which is stochastically dominated by a random variable X with E|X| < ∞. Xn,j = Yn,j – EYn,j + Zn,j – EZn,j, 1 ≤ j ≤ kn, n ≥ 1 It follows from Lemma 2.1 that {Yn,j, 1 ≤ j ≤ kn, n ≥ 1} is an array of rowwise PNQD random variables. Remark 3.1 One of the reviewers so kindly called to our attention the article by Ordóñez Cabrera and Volodin [9] and suggested that we should provide a comparison between Theorem 3.1 above and Theorem 1 of that article Both theorems are in the same spirit in that they both establish mean convergence for weighted averages from an array of rowwise PNQD mean 0 random variables. For n ≥ 1, since the set of random variables {Yn,j, 1 ≤ j ≤ kn} is PNQD by Lemma 2.1,
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