Convergence of two-dimensional branching recursions

The asymptotic distribution of branching type recursions (Ln) of the form Ln=d A Ln−1+B L̄n−1 is investigated in the two-dimensional case. Here L̄n−1 is an independent copy of Ln−1 and A,B are random matrices jointly independent of Ln−1,L̄n−1. The asymptotics of Ln after normalization are derived by a contraction method. The limiting distribution is characterized by a fixed point equation. The assumptions of the convergence theorem are checked in some examples using eigenvalue decompositions and computer algebra.