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In this paper, some new results on complete convergence and complete moment convergence for sequences of pairwise negatively quadrant dependent random variables are presented. These results improve the corresponding theorems of S. X. Gan, P. Y. Chen (2008) and H. Y. Liang, C. Su (1999).

Let $B^{H_i,K_i}=\{B_t^{H_i,K_i},t\ge 0\}$, $i=1,2$ be two independent, $d$-dimensional bifractional Brownian motions with respective indices $H_i\in (0,1)$ and $K_i\in (0,1]$. Assume $d\ge 2$. One of the main motivations of this paper is to investigate smoothness of the collision local time $${\ell }_T={\int }_0^T\delta (B_s^{H_1,K_1}-B_s^{H_2,K_2})\mathrm{d}s,T>0,$$ where $\delta$ denotes the Dirac delta function. By an elementary method we show that ${\ell }_T$ is smooth in the sense of Meyer-Watanabe if and only if $min\{H_1{K}_1,H_2{K}_2\}<1/(d+2)$.