Convergence of two-dimensional branching recursions

The asymptotic distribution of branching type recursions (L_n) of the form is investigated in the two-dimensional case. Here is an independent copy of L_n−1 and A,B are random matrices jointly independent of . The asymptotics of L_n 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.