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

Abstract

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 $\sum\nolimits_{j = 1}^{m(n)} {{X_{n,j}}/{b_n} \to 0}$ completely and for ${\max _{1 \leqslant k \leqslant m(n)}}|\sum\nolimits_{j = 1}^k {{X_{n,j}}|/{b_n} \to 0}$ 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 $\{ \{ X_{n,j}^*,1 \leqslant j \leqslant m(n)\} ,n \geqslant 1\}$ arising from a sequence of i.i.d. random variables {X _n , n ⩾ 1}.