  Let $B^{H,K}=(B^{H,K}_{t}, t\geq 0)$ be a bifractional Brownian motion with
two parameters $H\in (0,1)$ and $K\in(0,1]$. The main result of this paper is
that the increment process generated by the bifractional Brownian motion
$(B^{H,K}_{h+t} -B^{H,K}_{h}, t\geq 0)$ converges when $h\to \infty$ to
$(2^{(1-K)/{2}}B^{HK}_{t}, t\geq 0)$, where $(B^{HK}_{t}, t\geq 0)$ is the
fractional Brownian motion with Hurst index $HK$. We also study the behavior of
the noise associated to the bifractional Brownian motion and limit theorems to
$B^{H,K}$.
