Journal of Mathematics of Kyoto University
- J. Math. Kyoto Univ.
- Volume 49, Number 2 (2009), 267-286.
Convergence of dependent walks in a random scenery to fBm-local time fractional stable motions
It is classical to approximate the distribution of fractional Brownian motion by a renormalized sum $ S_n $ of dependent Gaussian random variables. In this paper we consider such a walk $ Z_n $ that collects random rewards $ \xi_j $ for $ j \in \mathbb Z$, when the ceiling of the walk $ S_n $ 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 increments limits.
J. Math. Kyoto Univ., Volume 49, Number 2 (2009), 267-286.
First available in Project Euclid: 22 October 2009
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Cohen, Serge; Dombry, Clément. Convergence of dependent walks in a random scenery to fBm-local time fractional stable motions. J. Math. Kyoto Univ. 49 (2009), no. 2, 267--286. doi:10.1215/kjm/1256219156. https://projecteuclid.org/euclid.kjm/1256219156