  Let $X$ be a random vector in $\rd$ with a regularly varying tail. We
consider two transformations $\|X\|f(\frac{X}{\|X\|})$, $f: \sd\to\sd$, and
$Xf(\frac{X}{\|X\|})$, $f: \sd\to \mathbb{R}_+$. Some sufficient conditions for
preserving the property of regularity of the tail for this kind of
transformations are given.
