Strong approximation of spatial random walk in random scenery. (Q1877519)

A random walk in random scenery is the process \(X(T)=\sum _{t=0}^{T}Y(S_t)\), \(T=0,1,2,\ldots \), where \(Y(x)\), \(x\in Z^d\), is an i.i.d.\ array and \(S_t\) is a random walk in \(Z^d\), cf. \textit{H. Kesten} and \textit{F. Spitzer} [Z. Wahrscheinlichkeitstheorie Verw. Geb. 50, 5--25 (1979; Zbl 0396.60037)]. For \(d\geq 3\) a random walk in random scenery is shown to be strongly approximated by a constant multiple of a real-valued Wiener process. The multiplying constant and orders of approximation are identified.