Recurrent random walks in nonnegative matrices. II (Q1326252)

We continue the study undertaken in our earlier paper [ibid. 91, No. 3/4, 297-306 (1992; Zbl 0739.60006)]. One of the main results here can be described as follows. Let \(X_ 0, X_ 1,\dots\) be a sequence of i.i.d. random affine maps from \((R^ +)^ d\) into itself. Let us write: \(W_ n \equiv X_ n X_{n-1} \dots X_ 0\) and \(Z_ n \equiv X_ 0 X_ 1 \dots X_ n\), where ``composition'' of maps is the rule of multiplication. By the attractor \({\mathcal A} (u)\), \(u \in(R^ +)^ d\), we mean the set \({\mathcal A} (u)=\{y \in (R^ +)^ d:P(W_ nu \in N\) i.o.) \(>0\) for every open \(N\) containing \(y\}\). It is shown that the attractor \({\mathcal A} (u)\), under mild conditions, is the support of a stationary probability measure, when the random walk \((Z_ n)\) has at least one recurrent state.