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Keywords:
array of rowwise pairwise negative quadrant dependent random variables; complete convergence; dependent bootstrap; sequence of i.i.d.\ random variables
array of rowwise pairwise negative quadrant dependent random variables; complete convergence; dependent bootstrap; sequence of i.i.d.\ random variables
Summary:
Let $\{X_{n,j}, 1\leq j\leq m(n), n\geq 1\}$ be an array of rowwise pairwise negative quadrant dependent mean 0 random variables and let $0<b_n\rightarrow \infty $. Conditions are given for $\sum \nolimits _{j=1}^{m(n)}X_{n,j}/b_n\rightarrow 0$ completely and for $\max \nolimits _{1\leq k\leq m(n)}\Bigl |\sum \nolimits _{j=1}^kX_{n,j}\Big |/b_n\rightarrow 0$ completely. As an application of these results, we obtain a complete convergence theorem for the row sums $\sum \nolimits _{j=1}^{m(n)}X_{n,j}^*$ of the dependent bootstrap samples $\{\{X_{n,j}^*, 1\leq j\leq m(n)\}, n\geq 1\}$ arising from a sequence of i.i.d.\ random variables $\{X_n, n\geq 1\}$.
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