Limiting behaviours for weighted sums of $m$-WOD random variables under integrability assumptions

In this work, the $L_r$-convergence for weighted sums of $R$-$h$-integrable $m$-WOD random variables, the complete convergence, complete moment convergence, and complete $f$-moment convergence for weighted sums of $SR$-$h$-integrable $m$-WOD random variables are established, which improve and generalize some previously obtained results in the literature. As an application, the weak consistency of the wavelet estimator in nonparametric regression models is obtained under very general assumptions.