Bifractional Brownian motion for $H>1$ and $2HK\le 1$

Bifractional Brownian motion on ${\mathbb{R}}_{+}$ is a two parameter centered Gaussian process with covariance function: $$R_{H,K}(t,s)=\frac{1}{2^K}({(t^{2H}+s^{2H})}^K-\mathrm{\mid }t-s{\mathrm{\mid }}^{2HK}),s,t\ge 0.$$ This process has been originally introduced by Houdr'e and Villa (2003) for the range of parameters $H\in (0,1]$ and $K\in (0,1]$\mathrm{.}Sincethen,therangeofparameters,forwhich$R_{H,K}$ is known to be nonnegative definite has been somewhat extended, but the full range is still not known. We give an elementary proof that $R_{H,K}$ is nonnegative definite for parameters $H,K$ satisfying $H>1$ and $0<2HK\le 1$. We show that $R_{H,K}$ can be decomposed into a sum of two nonnegative definite functions. As a side product we obtain a decomposition of the fractional Brownian motion with Hurst parameter $H<\frac{1}{2}$ into a sum of time rescaled Brownian motion and another independent self-similar Gaussian process. We also discuss some simple properties of bifractional Brownian motion with $H>1$.