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

Let $$\{D(x), x\geq 0\}$$ be a strictly stable subordinator with index $$\beta \in (0,1)$$, and $$\{E(t), t\geq 0\}$$ be the corresponding hitting time process. The authors prove (Proposition 3.1) that the process $$E(t)$$ is self-similar with exponent $$\beta$$. They also point out (Corollaries 3.1–3.3, Remark 3.1) some distributional and moment properties of $$E(t)$$ and verify that $$E(t)$$ has neither stationary nor independent increments.
Let $$\{N(t), t\geq 0\}$$ be a standard renewal process whose inter-arrival time belongs to the domain of attraction of a positive strictly stable law; $$Y_1, Y_2,\ldots$$ be i.i.d. random vectors, which are independent of $$N(t)$$. The process $$X_t:=Y_1+\ldots+Y_{N(t)}, t\geq 0$$, 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 $$\{M(t):=A(E(t)), t\geq 0\}$$. 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 $$\beta E$$, 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 $$M(t)$$ 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 Other physical applications of random processes
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### References:

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