Multivariate R-O Varying Measures Part I: Uniform Bounds

Abstract

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.