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Limit theorems for continuous-time random walks with infinite mean waiting times. (English) Zbl 1065.60042

Let $\left\{D\left(x\right),x\ge 0\right\}$ be a strictly stable subordinator with index $\beta \in \left(0,1\right)$, and $\left\{E\left(t\right),t\ge 0\right\}$ be the corresponding hitting time process. The authors prove (Proposition 3.1) that the process $E\left(t\right)$ is self-similar with exponent $\beta$. They also point out (Corollaries 3.1–3.3, Remark 3.1) some distributional and moment properties of $E\left(t\right)$ and verify that $E\left(t\right)$ has neither stationary nor independent increments.

Let $\left\{N\left(t\right),t\ge 0\right\}$ be a standard renewal process whose inter-arrival time belongs to the domain of attraction of a positive strictly stable law; ${Y}_{1},{Y}_{2},...$ be i.i.d. random vectors, which are independent of $N\left(t\right)$. The process ${X}_{t}:={Y}_{1}+...+{Y}_{N\left(t\right)},t\ge 0$, is called a continuous-time random walk. Let $A\left(t\right)$ be an operator Lévy motion. The authors prove (Theorem 3.2 and Corollary 3.4) that all finite-dimensional distributions of properly scaled and normalized process $N\left(t\right)$ converge to finite-dimensional distributions of $E\left(t\right)$. The convergence is also established in the Skorokhod space equipped with the ${J}_{1}$-topology. Theorem 4.2 contains a similar result for ${X}_{t}$. There the Skorokhod space is equipped with the ${M}_{1}$-topology, and the limiting process is $\left\{M\left(t\right):=A\left(E\left(t\right)\right),t\ge 0\right\}$. Further (Corollaries 4.1–4.3, Theorem 4.3) the authors investigate properties of the process $M\left(t\right)$. In particular, they check that the process is operator self-similar with exponent $\beta E$, where $E$ is a matrix with real entries, but not operator stable; it does not have stationary increments. Finally, Theorem 5.1 states that the density of $M\left(t\right)$ solves a fractional kinetic equation which is a generalization of a fractional partial differential equation for Hamiltonian chaos.

##### MSC:
 60G50 Sums of independent random variables; random walks 60K40 Physical applications of random processes