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Let 0

For bootstrap sample sums resulting from a sequence of random variables { X n , n ≥ 1 } , a very general central limit theorem is established. The random variables { X n , n ≥ 1 } do not need to be independent or identically distributed or to be of any particular dependence structure. Furthermore, no conditions, including moment conditions, are imposed in general on the marginal distributions of the { X n , n ≥ 1 } . As a special case of the main result, a result of Liu (1988) concerning independent but non-identically distributed { X n , n ≥ 1 } is extended to a larger class of parent sequences. A version of the main result is also presented wherein the bootstrap sample sums have random bootstrap sample sizes. [ABSTRACT FROM AUTHOR]

For an array of rowwise pairwise negative quadrant dependent, mean 0 random variables, Chandra, Li, and Rosalsky provided conditions under which weighted averages converge in L1 to 0. The Chandra, Li, and Rosalsky result is extended to Lr convergence (1≤r<2) and is shown to hold under weaker conditions by applying a mean convergence result of Sung and an inequality of Adler, Rosalsky, and Taylor. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]