A Random Walk in a Quadratic Random Scenery

N. Guillotin-Plantard

2001, v.7, Issue 3, 419-434

ABSTRACT

Let $(X_{i})_{i\geq 1}$ be a sequence of independent and identically distributed $Z^{d}$-valued random vectors defined on a probability space $(\Omega,F,P)$ and $S_{n}=\sum_{i=1}^{n}X_{i}$ for $n\geq 1$. Let $(\xi_{\alpha})_{\alpha\in Z^{d}}$ be a sequence of independent $R^{m}$-valued random vectors defined on a probability space $(E,A,Q)$ and $$G(t)=\sup_{x\neq y}Q\{||\xi_{x}||\,||\xi_{y}||\geq t\}.$$ Let us denote $$Q_{n}=\sum_{i,j=0}^{n}a_{i,j}^{(n)}\xi_{S_{i}} \cdot\xi_{S_{j}}, n\geq 0.$$ We give conditions on the reals $a_{i,j}^{(n)}$ and the function $G$ for which the sequence $(Q_{n})_{n\geq 0}$ converges almost surely to 0.

Keywords: random walk,random scenery,quadratic forms,almost sure convergence