Linear growth in near-critical population-size dependent multitype Galton-Watson processes

The paper considers a multitype population-size-dependent branching process in discrete time. A process is considered to be near-critical if the mean matrices of offspring distributions approach the mean matrix of a critical process as the population size increases. The paper shows that if the second moments of offspring distributions stabilize as the population size increases, and the limiting variances are not too large in comparison with the deviation of the means from criticality, then the extinction probability is less than 1 and the process grows arithmetically fast, in the sense that any linear combination which is not orthogonal to the left eigenvector of the limiting mean matrix grows linearly to a limit distribution. The paper identifies cases when the limiting distribution is gamma. A result on transience of multidimensional Markov chains is also given.