Limits of sub-bifractional Brownian noises

Let $S^{H, K} = \{S^{H, K}_t, t\geq 0\}$ 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 $\left\{S^{H, K}_{h+t}-S^{H, K}_h, t\geq 0\right\}$ converges to $\left\{B^{HK}_t, t\geq 0\right\}$ as $h\rightarrow \infty$, where $\left\{B^{HK}_t, t\geq 0\right\}$ 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