Let be a strictly stable subordinator with index , and be the corresponding hitting time process. The authors prove (Proposition 3.1) that the process is self-similar with exponent . They also point out (Corollaries 3.1–3.3, Remark 3.1) some distributional and moment properties of and verify that has neither stationary nor independent increments.
Let be a standard renewal process whose inter-arrival time belongs to the domain of attraction of a positive strictly stable law; be i.i.d. random vectors, which are independent of . The process , is called a continuous-time random walk. Let 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 converge to finite-dimensional distributions of . The convergence is also established in the Skorokhod space equipped with the -topology. Theorem 4.2 contains a similar result for . There the Skorokhod space is equipped with the -topology, and the limiting process is . Further (Corollaries 4.1–4.3, Theorem 4.3) the authors investigate properties of the process . In particular, they check that the process is operator self-similar with exponent , where 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 solves a fractional kinetic equation which is a generalization of a fractional partial differential equation for Hamiltonian chaos.