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

Let {D(x),x0} be a strictly stable subordinator with index β(0,1), and {E(t),t0} be the corresponding hitting time process. The authors prove (Proposition 3.1) that the process E(t) is self-similar with exponent β. They also point out (Corollaries 3.1–3.3, Remark 3.1) some distributional and moment properties of E(t) and verify that E(t) has neither stationary nor independent increments.

Let {N(t),t0} 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(t). The process X t :=Y 1 +...+Y N(t) ,t0, is called a continuous-time random walk. Let A(t) 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(t) converge to finite-dimensional distributions of E(t). 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 {M(t):=A(E(t)),t0}. Further (Corollaries 4.1–4.3, Theorem 4.3) the authors investigate properties of the process M(t). In particular, they check that the process is operator self-similar with exponent βE, 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(t) solves a fractional kinetic equation which is a generalization of a fractional partial differential equation for Hamiltonian chaos.

60G50Sums of independent random variables; random walks
60K40Physical applications of random processes