  Let $B=\{(B_{t}^{1},..., B_{t}^{d}), t\geq 0\}$ be a $d$-dimensional
fractional Brownian motion with Hurst parameter $H$ and let $R_{t}=%
\sqrt{(B_{t}^{1})^{2}+... +(B_{t}^{d})^{2}}$ be the fractional Bessel process.
It\^{o}'s formula for the fractional Brownian motion leads to the equation $
R_{t}=\sum_{i=1}^{d}\int_{0}^{t}\frac{B_{s}^{i}}{R_{s}}%
dB_{s}^{i}+H(d-1)\int_{0}^{t}\frac{s^{2H-1}}{R_{s}}ds . $ In the Brownian
motion case ($H=1/2$), $X_{t}=\sum_{i=1}^{d}\int_{0}^{t} frac{B_{s}^{i}}{%
R_{s}}dB_{s}^{i}$ is a Brownian motion. In this paper it is shown that $X_{t}$
is \underbar{not} a fractional Brownian motion if $H\not=1/2$. We will study
some other properties of this stochastic process as well.
