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We consider the Galton-Watson branching process allowing immigration. We are dealing with the critical case, in which the immigration law has infinite mean and the offspring law have an infinite variance. An explicit-integral form of the generating function of a stationary measure for the process without immigration is found. We study the asymptotic properties of transition probabilities and their convergence to stationary measures in the case of processes with immigration, when the process is ergodic. And also we define a rate of speed of this convergence.

On asymptotic structure of critical Galton-Watson branching processes allowing immigration with infinite variance