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Convergence of dependent walks in a random scenery to fBm-local time fractional stable motions. (English) Zbl 1205.60082

After giving some motivations the authors explain their concept of a random walk in a random scenery, and they assume the elements of this scenery belonging to the normal domain of attraction of a stable symmetric distribution of index \(\beta \in (0, 2)\). Stated simply, a random walk in a random scenery is a cumulative sum process whose summands are drawn from the scenery, the order in which the summands are drawn is determined by the path of the random walk.
Best in the authors’ own words: We consider a walk \(Z_n\) that collects random rewards \(\xi_j\) (for \(j\) running through the integers) when the ceiling of the walk \(S_n\) (which is a renormalized sum of dependent Gaussian random variables) is located at \(j\). The random reward (or scenery) \(\xi_j\) is independent of the walk and with heavy tail. We show the convergence of the sum of independent copies of \(Z_n\) suitably renormalized to a stable motion with integral representation, whose kernel is the local time of a fractional Brownian motion (fBm). This work extends a previous work where the random walk \(S_n\) had independent increment limits.

MSC:

60G18 Self-similar stochastic processes
60G52 Stable stochastic processes
60K37 Processes in random environments
60F17 Functional limit theorems; invariance principles
60G22 Fractional processes, including fractional Brownian motion
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