Improved results on \(\mathcal {L}_{r}\)-convergence for weighted sums of dependent sequences

This work is inspired by an open problem raised by Thanh (RACSAM 118(Article ID 40):17, 2024. https://doi.org/10.1007/s13398-023-01540-5). Some improved results on the \(\mathcal {L}_r\) ${\mathcal{L}}_r$ convergence for weighted sums of m-WOD random variables are derived, which give a positive answer to the open problem in Thanh (2024). These results also improve and generalise some corresponding ones concerning the R-h-integrable random variables in the literature.