Stochastic vector difference equations with stationary coefficients

We give a unified presentation of stability results for stochastic vector difference equations Y_n+1 = A_n ⊗ Y_n ⊕ B_n based on various choices of binary operations ⊕ and ⊗, assuming that {(A_n, B_n), n ≥ 0} are stationary and ergodic. In the scalar case, under standard addition and multiplication, the key condition for stability is E[log |A₀|]<0. In the generalizations, the condition takes the form γ <0, where γ is the limit of a subadditive process associated with {A(n), n≥0}. Under this and mild additional conditions, the process { Y_n, n ≥ 0} has a unique finite stationary distribution to which it converges from all initial conditions The variants of standard matrix algebra we consider replace the operations + and x with (max,+), (max,x), (min,+), or (min,x). In each case, the appropriate stability condition parallels that for the standard recursions, involving certain subadditive limits. Since these limits are difficult to evaluate, we provide bounds, thus giving alternative, computable conditions for stability.