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Zbl 1179.60024
Meerschaert, Mark M.; Nane, Erkan; Xiao, Yimin
Correlated continuous time random walks.
(English)
[J] Stat. Probab. Lett. 79, No. 9, 1194-1202 (2009). ISSN 0167-7152

Let $(Z_j: j\in \bbfZ)$ and $(J_k: k\in \bbfN)$, $J_k>0$, be independent sequences of i.i.d. random variables, and $(N_t: t\geq 0)$ be a renewal process generated by the $J_k$'s. Assume that the laws of $Z_1$ and $J_1$ belong to the domains of attraction of a strictly $\alpha$-stable law, $\alpha\in (0,2]$, and of a $\beta$-stable law, $\beta\in (0,1)$, respectively. Define $Y_n:=\sum_{j=0}^\infty c_j Z_{n-j}$, $n\in \bbfZ$ for some real constants $c_j$ satisfying certain conditions which, among others, ensure the a.s. convergence of the latter series. \par Depending on summability properties of $c_j$'s, the paper under review proves four weak convergence results for, properly scaled, continuous time random walk $(Y_1+\ldots+Y_{N_t}: t\geq 0)$. In particular, it is shown that the set of possible limiting processes includes four distinct processes subordinated to an inverse $\beta$-stable subordinator (1) a stable subordinator (convergence holds in the $M_1$ topology on $D[0,\infty)$); (2) a linear fractional stable motion (LFSM) with a.s. continuous paths (convergence holds in the $J_1$ topology on $D[0,\infty)$); (3) a LFSM with a.s. unbounded paths on every interval of positive length (convergence of finite-dimensional distributions); (4) a fractional Brownian motion (convergence holds in the $J_1$ topology on $D[0,\infty)$). \par Notice that the first case bears resemblance with a classical situation when $(Y_n: n\in \bbfZ)$ are i.i.d.
[Aleksander Iksanov (Kiev)]
MSC 2000:
*60G50 Sums of independent random variables
60F17 Functional limit theorems
60G18 Self-similar processes
60G52 Stable processes

Keywords: continuous time random walk; domain of attraction of a stable law; functional limit theorem; long-range dependence

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