## Journal of Mathematics of Kyoto University

### Convergence of dependent walks in a random scenery to fBm-local time fractional stable motions

#### Abstract

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.

#### Article information

Source
J. Math. Kyoto Univ., Volume 49, Number 2 (2009), 267-286.

Dates
First available in Project Euclid: 22 October 2009

https://projecteuclid.org/euclid.kjm/1256219156

Digital Object Identifier
doi:10.1215/kjm/1256219156

Mathematical Reviews number (MathSciNet)
MR2571841

Zentralblatt MATH identifier
1205.60082

#### Citation

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