Convergence of two-dimensional branching recursions

摘要

The asymptotic distribution of branching typer recursions (L_n) of the form L_n (=d) A L_(n-1) + B(L-bar)_(n-1) is investigated in the two-dimensional case. Here (L-bar)_(n-1) is an independent copy of L_(n-1) and A, B are random matrices jointly independent of L_(n-1), (L-bar)_(n-1). 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.