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We investigate the asymptotic behaviour of homogeneous multidimensional Markov chains whose states have nonnegative integer components. We obtain growth rates for these models in a situation similar to the near-critical case for branching processes, provided that they converge to infinity with positive probability. Finally, the general theoretical results are applied to a class of controlled multitype branching process in which random control is allowed.doi:10.1239/jap/1143936250 fatcat:rwzfau2yo5a7zpvljd7wlqpsva