Let $S^{H,K}=\{S_t^{H,K},t\ge 0\}$ be the sub-bifractional Brownian motion (sbfBm) of dimension 1, with indices $H\in (0,1)$ and $K\in (0,1].$ We primarily prove that the increment process generated by the sbfBm $\left\{S_{h+t}^{H,K}-S_h^{H,K},t\ge 0\right\}$ converges to $\left\{B_t^{HK},t\ge 0\right\}$ as $h\to \mathrm{\infty }$, where $\left\{B_t^{HK},t\ge 0\right\}$ is the fractional Brownian motion with Hurst index $HK$. Moreover, we study the behavior of the noise associated to the sbfBm and limit theorems to $S^{H,K}$ and the behavior of the tangent process of sbfBm.