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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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