A strong invariance principle for two-dimensional random walk in random scenery. (Q1879529)

Let \(\{\mathbf S_n, n\geq 0\}\) with \(\mathbf S_0 = \mathbf 0 \) be a symmetric random walk on the lattice \(\mathbb Z^2;\) let \(\{Y(\mathbf x), \mathbf x \in \mathbb Z^2 \}\) be a collection of i.i.d. random variables with zero mean and a positive finite variance. Let \(\{Z_n ,n\geq 0\} \) be a random walk in random scenery process such that \( Z(n) = \sum _{j=0}^n Y(\mathbf S_j). \) A strong invariance principle for \(\{Z(n), n\geq 0\}\) is presented, from which weak convergence and Strassen- and Chung-type laws of the iterated logarithm follows as consequences.