### Limit theorems for one and two-dimensional random walks in random scenery

Fabienne Castell, Nadine Guillotin-Plantard, Françoise Pène (2013)

Annales de l'I.H.P. Probabilités et statistiques

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Random walks in random scenery are processes defined by ${Z}_{n}:={\sum}_{k=1}^{n}{\xi}_{{X}_{1}+\cdots +{X}_{k}}$, where $({X}_{k},k\ge 1)$ and $({\xi}_{y},y\in {\mathbb{Z}}^{d})$ are two independent sequences of i.i.d. random variables with values in ${\mathbb{Z}}^{d}$ and $\mathbb{R}$ respectively. We suppose that the distributions of ${X}_{1}$ and ${\xi}_{0}$ belong to the normal basin of attraction of stable distribution of index $\alpha \in (0,2]$ and $\beta \in (0,2]$. When $d=1$ and $\alpha \ne 1$, a functional limit theorem has been established in ( (1979) 5–25) and a local limit theorem in (To appear). In this paper, we establish the convergence in distribution...