STOCHASTIC VECTOR DIFFERENCE-EQUATIONS WITH STATIONARY COEFFICIENTS

We give a unified presentation of stability results for stochastic vec tor difference equations Y-n+1 = A(n)X Y-n+B-n based on various choice s of binary operations + and X, assuming that {(A(n), B-n), n greater than or equal to O} are stationary and ergodic. In the scalar case, un der standard addition and multiplication, the key condition for stabil ity is E[log\A(0)\]<O. In the generalizations, the condition takes the form gamma<0, where gamma is the limit of a subadditive process assoc iated with {A(n), n greater than or equal to O}. Under this and mild a dditional conditions, the process {Y-n, n greater than or equal to O} has a unique finite stationary distribution to which it converges from all initial conditions. The variants of standard matrix algebra we co nsider replace the operations + and x with (max,+), (max, x), (min,+), or (min, x). In each case, the appropriate stability condition parall els that for the standard recursions, involving certain subadditive li mits. Since these limits are difficult to evaluate, we provide bounds, thus giving alternative, computable conditions for stability.