More Exact Statements of Several Theorems in the Theory of Branching Processes

Theorems proved by A. N. Kolmogorov [2], A. M. Yaglom [3] and B. A. Sevastyanov [1] on regular branching processes with particles of one type homogeneous in time (parameter $t$ being continuous) are made more exact. It is possible to do away with the requirement that the second and third factorial moments be finite in the integral limit theorems. For processes with $a=f'(1)\ne0$, the form of limit theorems is the same as before. For processes with $a=0$, it is necessary to require in addition that the generating function determining the branch process be properly variable in the sense of Karamata [4] with the index $\gamma=1+\alpha>1$. In this case, a new class of distributions appears as the limit laws. The Laplace transforms for these distributions have a very simple form
$$\psi(s)=1-s(1+s^\alpha)^{-1/\alpha},\quad0<\alpha\leq1.$$
The limit distributions of this class $S_\alpha(y)$ may be expressed by means of the densities of stable distributions $p(x,\alpha,\beta)$ with the parameters $0<\alpha<1$ and $\beta=1$ as follows:
$$S_\alpha(y)=\begin{cases}\displaystyle1-\dfrac{\alpha}{\Gamma(1/\alpha)}\int_0^\infty e^{-(y/u)^\alpha}p(u,\alpha,1)\dfrac{du}{u}&\quad \text{ for }0<\alpha<1,\\1-e^{-y}&\quad\text{ for }\alpha=1.\end{cases}$$
The asymptotic behavior of the probability $Q(t)$ that the process will degenerate and the probabilities $P_n(t)$ that at moment $t$ there will be just $n$ particles $(n\geq1)$ is also made more exact.