On asymptotic structure of continuous-time Markov branching processes allowing immigration without higher-order moments

We consider a continuous-time Markov branching process allowing immigration. Our main analytical tool is the slow variation (or more general, a regular variation) conception in the sense of Karamata. The slow variation property arises in many issues, but it usually remains rather hidden. For example, denoting by () the perimeter of an equilateral polygon with sides inscribed in a circle with a diameter of length , one can check that the function () := ()/ converges to in the sense of Archimedes, but it slowly varies at infinity in the sense of Karamata. In fact, it is known that () = sin (/) and then it follows ()/() 1 as for each > 0. Thus, () is so slowly approaching that it can be suspected that " is not quite constant".

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