*(English)*Zbl 1065.60042

Let $\left\{D\right(x),x\ge 0\}$ be a strictly stable subordinator with index $\beta \in (0,1)$, and $\left\{E\right(t),t\ge 0\}$ 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 $\left\{N\right(t),t\ge 0\}$ 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 $\left\{M\right(t):=A(E\left(t\right)),t\ge 0\}$. 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.