EUDML

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 $0 < b_{n} \to \infty$ . Conditions are given for $\sum_{j = 1}^{m (n)} X_{n, j} / b_{n} \to 0$ completely and for ${max}_{1 \leq k \leq m (n)} | \sum_{j = 1}^{k} X_{n, j} | / b_{n} \to 0$ completely. As an application of these results, we obtain a complete convergence theorem for the row sums $\sum_{j = 1}^{m (n)} X_{n, j}^{*}$ 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 ${X_{n}, n \geq 1}$ .

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 ( $m = 1, 2, \dots$ ). For a sequence of $m$ -linearly negative quadrant dependent random variables ${X_{n}, n \geq 1}$ and $1 < p < 2$ (resp. $1 \leq p < 2$ ), conditions are provided under which $n^{- 1 / p} \sum_{k = 1}^{n} (X_{k} - E X_{k}) \to 0$ in $L^{1}$ (resp. in $L^{p}$ ). Moreover, for $1 \leq p < 2$ , conditions are provided under which $n^{- 1 / p} \sum_{k = 1}^{n} (X_{k} - E X_{k})$ converges completely to $0$ . The current work extends some results of Pyke and Root (1968) and it extends and improves some...

A correction to “Some mean convergence and complete convergence theorems for sequences of $m$ -linearly negative quadrant dependent random variables”

Yongfeng Wu; Andrew Rosalsky; Andrei Volodin — 2017

Applications of Mathematics

The authors provide a correction to “Some mean convergence and complete convergence theorems for sequences of $m$ -linearly negative quadrant dependent random variables”.

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A generalization of the global limit theorems of R. P. Agnew.

Addendum: A generalization of the global limit theorems of R. P. Agnew.

A survey of limit laws for bootstrapped sums.

On the weak law of large numbers for normed weighted sums of i.i.d. random variables.

Precise lim sup behavior of probabilities of large deviations for sums of i.i.d. random variables.

On convergence of series of random elements via maximal moment relations with applications to martingale convergence and to convergence of series with $p$ -orthogonal summands.

On convergence of series of random elements via maximal moment relations with applications to martingale convergence and to convergence of series with $p$ -orthogonal summands. Correction.

On the asymptotic probability for the deviations of dependent bootstrap means from the sample mean.

Strong laws of large numbers for weighted sums of random elements in normed linear spaces.

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

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

A correction to “Some mean convergence and complete convergence theorems for sequences of $m$ -linearly negative quadrant dependent random variables”