A finite Borel measure $\mu$ on $\mathbb R^d$ is
called R-O varying with index $F$ if there exist a
$\operatorname{GL}(\mathbb R^d)$-valued function $f$
varying regularly with index $(-F)$, an increasing
function $k : (0,\infty)\to (0,\infty)$ with
$k(t)\to\infty$ and $k(t+1)/k(t)\to c\geq 1$ as
$t\to\infty$, and a $\sigma$-finite measure $\phi$ on
$\mathbb R^d\setminus\{0\}$ such that
\[k(t)\cdot(f(k(t))\mu)\to\phi\quad
\text{as $t\to\infty$.}\]
R-O varying measures generalize regularly varying
measures introduced by Meerschaert (see
M.~M.~Meerschaert, `Regular variation in $\mathbb R^k$',
{\it Proc.\ Amer.\ Math.\ Soc.} 102 (1988) 341--348)
and have numerous applications in limit theorems for
probability measures.
For an R-O varying measure $\mu$ and $-\infty<\infty$
let
\begin{equation*}
\begin{split}
V_a(t,\theta) &= \int_{|\langle x,\theta\rangle|>t}
|\langle x,\theta\rangle|^a\,d\mu(x),\\
U_b(t,\theta) &= \int_{|\langle x,\theta\rangle|\leq t}
|\langle x,\theta\rangle|^b\,d\mu(x)
\end{split}
\end{equation*}
denote the tail- and truncated moment functions of $\mu$
in the direction $\|\theta\|=1$. The purpose of this paper
is to show that R-O variation of a measure implies sharp
bounds on the growth rate of the tail- and truncated
moment functions depending on the real parts of the
eigenvalues of the index $F$ along a compact set of
directions. Furthermore, bounds on the ratio of these
functions for certain values of $a$ and $b$ are obtained. 1991 Mathematics Subject Classification: 60B10, 28C15.
