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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. (English) Zbl 1363.60038
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 \(\{X_n, n\geq 1\}\).
MSC:
60F15 Strong limit theorems
62F40 Bootstrap, jackknife and other resampling methods
62G09 Nonparametric statistical resampling methods
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