Strong laws of large numbers for arrays of row-wise extended negatively dependent random variables with applications

João Lita da Silva

Journal of Nonparametric Statistics, 2020, vol. 32, issue 1, 20-41

Abstract: The main purpose of this paper is to obtain strong laws of large numbers for arrays or weighted sums of random variables under a scenario of dependence. Namely, for triangular arrays $\{X_{n,j},\ 1 \leqslant j \leqslant n,\ n \geqslant 1\} ${Xn,j, 1⩽j⩽n, n⩾1} of row-wise extended negatively dependent random variables weakly mean dominated by a random variable $X \in \mathscr {L}_{1} $X∈L1 and sequences $\{b_{n}\} ${bn} of positive constants, conditions are given to ensure $\sum _{j=1}^{n} \left (X_{n,j} - \mathbb {E}\, X_{n,j} \right )/b_{n} \overset {\hbox{a.s.}}{\longrightarrow } 0 $∑j=1nXn,j−EXn,j/bn⟶a.s.0. Our statements allow us to establish strong consistency of general nonparametric estimates in a nonparametric regression model having fixed design points.

Date: 2020
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DOI: 10.1080/10485252.2019.1688326

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