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

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

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Zolotarev, V. M.
Copyright Year	1957
Abstract	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 $a = f'(1) \ne 0$, 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 \[ \psi (s) = 1 - s(1 + s^\alpha )^{ - 1/ \alpha } ,\quad 0 < \alpha \leqq 1. \] 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: \[S_\alpha (y) = \left\{ \begin{gathered} 1 - \frac{\alpha }{{\Gamma (1/ \alpha )}}\int_0^\infty {e^{ - (y/u)^\alpha }} p(u,\alpha ,1)\frac{{du}}{u}\quad \text{for }0 < \alpha < 1, \hfill \\ \qquad\,\,\quad 1 - e^{ - y} \qquad \qquad\ \qquad \qquad\qquad\text{for }\alpha = 1. \hfill \\ \end{gathered} \right. \] 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 $(n \geqq 1)$ is also made more exact.
Starting Page	245
Ending Page	253
Page Count	9
File Format	PDF
ISSN	0040585X
DOI	10.1137/1102016
e-ISSN	10957219
Journal	Theory of Probability & Its Applications (TPRBAU)
Issue Number	2
Volume Number	2
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2006-07-28
Access Restriction	Subscribed
Content Type	Text
Resource Type	Article
Subject	Statistics and Probability Statistics, Probability and Uncertainty

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