  We characterize recurrence and transience of nonnegative multivariate
autoregressive processes of order one with random contractive coefficient
matrix, of subcritical multitype Galton-Watson branching processes in random
environment with immigration, and of the related max-autoregressive processes
and general random exchange processes. Our criterion is given in terms of the
maximal Lyapunov exponent of the coefficient matrix and the cumulative
distribution function of the innovation/immigration component.
