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Complete convergence theorems for normed row sums from an array of rowwise pairwise negative quadrant dependent random variables with application to the dependent bootstrap

Andrew Rosalsky;  Yongfeng Wu — 2015

Applications of Mathematics

Let { X n , j , 1 ≤ j ≤ m ( n ) , n ≥ 1 } be an array of rowwise pairwise negative quadrant dependent mean 0 random variables and let 0 < b n → ∞ . Conditions are given for ∑ j = 1 m ( n ) X n , j / b n → 0 completely and for max 1 ≤ k ≤ m ( n ) | ∑ j = 1 k X n , j | / b n → 0 completely. As an application of these results, we obtain a complete convergence theorem for the row sums ∑ j = 1 m ( n ) X n , j * of the dependent bootstrap samples { { X n , j * , 1 ≤ j ≤ m ( n ) } , n ≥ 1 } arising from a sequence of i.i.d. random variables { X n , n ≥ 1 } .

Some mean convergence and complete convergence theorems for sequences of m -linearly negative quadrant dependent random variables

Yongfeng Wu;  Andrew Rosalsky;  Andrei Volodin — 2013

Applications of Mathematics

The structure of linearly negative quadrant dependent random variables is extended by introducing the structure of m -linearly negative quadrant dependent random variables ( m = 1 , 2 , ⋯ ). For a sequence of m -linearly negative quadrant dependent random variables { X n , n ≥ 1 } and 1 < p < 2 (resp. 1 ≤ p < 2 ), conditions are provided under which n - 1 / p ∑ k = 1 n ( X k - E X k ) → 0 in L 1 (resp. in L p ). Moreover, for 1 ≤ p < 2 , conditions are provided under which n - 1 / p ∑ k = 1 n ( X k - E X k ) converges completely to 0 . The current work extends some results of Pyke and Root (1968) and it extends and improves some...

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