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The main results obtained from 1968 to 1983 in the theory of Markov branching processes and processes with transformations depending on the age of particles are reflected in this article. Along with the traditional sections (integral and local theorems, stationary measures), the survey includes sections devoted to statistics of branching processes. The bibliography contains mainly works reviewed in RZh Matematika.Translated from Itogi Nauki i Tekhniki, Seriya Teoriya Veroyatnostei, Matematicheskaya Stati tika, Teoreticheskaya Kibernetika, Vol. 23, pp. 3–67, 1985. 相似文献
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V. A. Vatutin 《Siberian Advances in Mathematics》2011,21(1):42-72
For multitype branching processes with immigration evolving in a random environment and producing a final product, we find
the tail distribution of the size of the final product accumulated in the process for a life period. Using this result, we investigate the tail distributions of the busy periods of the queueing polling systems
of branching type with random service disciplines and random positive switch-over times. 相似文献
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Some Limit Theorems for a Particle System of Single Point Catalytic Branching Random Walks 总被引:2,自引:0,他引:2
Vladimir VATUTIN Jie XIONG 《数学学报(英文版)》2007,23(6):997-1012
We study the scaling limit for a catalytic branching particle system whose particles perform random walks on Z and can branch at 0 only. Varying the initial (finite) number of particles, we get for this system different limiting distributions. To be more specific, suppose that initially there are n^β particles and consider the scaled process Zt^n(·) = Znt(√n·), where Zt is the measure-valued process 1 and to a representing the original particle system. We prove that Ztn converges to 0 when β 〈1/4 and to a nondegenerate discrete distribution when β=1/4.In addition,if 1/4〈β〈1/2 then n-^(2β-1/2)Zt^n converges to a random limit,while if β 〉21then n^-βZtn converges to a deterministic limit. 相似文献
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V. A. Vatutin E. E. Dyakonova S. Sagitov 《Proceedings of the Steklov Institute of Mathematics》2013,282(1):220-242
This review paper presents the known results on the asymptotics of the survival probability and limit theorems conditioned on survival of critical and subcritical branching processes in independent and identically distributed random environments. This is a natural generalization of the time-inhomogeneous branching processes. The key assumptions of the family of population models in question are nonoverlapping generations and discrete time. The reader should be aware of the fact that there are many very interesting papers covering other issues in the theory of branching processes in random environments which are not mentioned here. 相似文献
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Let S 0 = 0, {S n , n ≥ 1} be a random walk generated by a sequence of i.i.d. random variables X 1, X 2, . . . and let $\tau ^{-}={\rm min} \{ n \geq 1:S_{n}\leq 0 \}$ and $\tau ^{+}={\rm min}\{n\geq1:S_{n} > 0\} $ . Assuming that the distribution of X 1 belongs to the domain of attraction of an α-stable law we study the asymptotic behavior, as ${n\rightarrow \infty }$ , of the local probabilities ${\bf P}{(\tau ^{\pm }=n)}$ and prove the Gnedenko and Stone type conditional local limit theorems for the probabilities ${\bf P}{(S_{n} \in [x,x+\Delta )|\tau^{-} > n)}$ with fixed Δ and ${x=x(n)\in (0,\infty )}$ . 相似文献
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Klaus?FleischmannEmail author Vladimir?A.?Vatutin 《Journal of Theoretical Probability》2005,18(4):719-755
Consider a system of particles which move in Rd according to a symmetric α-stable motion, have a lifetime distribution of finite mean, and branch with an offspring law of index 1+β. In case of the critical dimension d=α/β the phenomenon of multi-scale clustering occurs. This is expressed in an fdd scaling limit theorem, where initially we start
with an increasing localized population or with an increasing homogeneous Poissonian population. The limit state is uniform,
but its intensity varies in line with the scaling index according to a continuous-state branching process of index 1+β. Our result generalizes the case α=2 of Brownian particles of Klenke (1998), where p.d.e. methods had been used which are not available in the present setting.
Supported in part by the DFG.
Supported in part by the grants RFBR 02-01-00266 and Russian Scientific School 1758.2003.1. 相似文献
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An integral test (Theorem 5) is established for the dichotomy concerning local extinction and survival (even persistence)
at late times for critical multitype spatially homogeneous branching particle systems in continuous time. Our conditions on
the branching mechanism are close to the ones known from “classical” processes without motion component. This generalizes
and complements results of López-Mimbela and Wakolbinger [LMW96] and others. Our approach is based on some genealogical tree
analysis combined with the study of the long-term behavior of L
1-norms of solutions of related systems of reaction-“diffusion” equations, which is perhaps also of some independent interest.
Received: 13 August 1997 / Revised version: 12 May 1998 / Published online: 14 February 2000 相似文献
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