Multivariate R-O varying measures. I: Uniform bounds (Q2766365)

A finite Borel measure \(\mu\) on \(\mathbb{R}^d\) is called R-O varying with index \(F\) if there exist a \(\text{GL}(\mathbb{R}^d)\)-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 NEWLINE\[NEWLINEk(t)\cdot (f(k(t))\mu)\to \phi\quad\text{as }t\to\infty.NEWLINE\]NEWLINE R-O varying measures generalize regularly varying measures introduced by \textit{M. M. Meerschaert} [``Regular variation in \(\mathbb{R}^k\)'', Proc. Am. Math. Soc. 102, No. 2, 341-348 (1988; Zbl 0648.26006)] and have numerous applications in limit theorems for probability measures. For an R-O varying measure \(\mu\) and \(-\infty< a< b<\infty\) let NEWLINE\[NEWLINEV_a(t,\theta)= \int_{|\langle x,\theta\rangle|> t}|\langle x,\theta\rangle|^a d\mu(x),\;U_b(t,\theta)= \int_{|\langle x,\theta\rangle|\leq t}|\langle x,\theta\rangle|^b d\mu(x)NEWLINE\]NEWLINE denote, respectively, 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 \(a\), \(b\) are obtained.