An extension on the rate of complete moment convergence for weighted sums of weakly dependent random variables

The authors study the convergence rate of complete moment convergence for weighted sums of weakly dependent random variables without assumptions of identical distribution. Under the moment condition of $E{\left|X\right|}^{\alpha }/{\left(\log \left(1+\left|X\right|\right)\right)}^{\alpha /\gamma -1}<\mathrm{\infty }$ for $0<\gamma <\alpha$ with $1<\alpha \le 2$, we establish the complete $\alpha$-th moment convergence theorem for weighted sums of weakly dependent cases, which improves and extends the related known results in the literature.