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Zbl 1065.60042
Meerschaert, Mark M.; Scheffler, Hans-Peter
Limit theorems for continuous-time random walks with infinite mean waiting times.
(English)
[J] J. Appl. Probab. 41, No. 3, 623-638 (2004). ISSN 0021-9002

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.
[Aleksander Iksanov (Kiev)]
MSC 2000:
*60G50 Sums of independent random variables
60K40 Physical appl. of random processes

Keywords: operator self-similar process; continuous-time random walk

Cited in: Zbl 1186.60079 Zbl 1147.60025

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