A continuous-time random walk is a simple random walk subordinated to a renewal process used in physics to model anomalous diffusion. In this paper we show that, when the time between renewals has infinite mean, the scaling limit is an operator Lévy motion subordinated to the hitting time process of a classical stable subordinator. Density functions for the limit process solve a fractional Cauchy problem, the generalization of a fractional partial differential equation for Hamiltonian chaos. We also establish a functional limit theorem for random walks with jumps in the strict generalized domain of attraction of a full operator stable law, which is of some independent interest.

Continuous-time random walks incorporate a random waiting time between random jumps. They are used in physics to model particle motion. A physically realistic rescaling uses two different time scales for the mean waiting time and the deviation from the mean. This paper derives the scaling limits for such processes. These limit processes are governed by fractional partial differential equations that may be useful in physics. A transfer theorem for weak convergence of finite-dimensional distributions of stochastic processes is also obtained.

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 thesefunctions for certain values of $a$ and $b$ are obtained. 1991 Mathematics Subject Classification: 60B10, 28C15.