Limits of bifractional Brownian noises

Abstract

Let $B^{H,K}=(B_t^{H,K},t\ge 0)$ 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 $(B_{h+t}^{H,K}-B_h^{H,K},t\ge 0)$ converges when $h\to \infty$ to $(2^{(1-K)/2}{B}_t^{HK},t\ge 0)$, where $(B_t^{HK},t\ge 0)$ 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 $B^{H,K}$.