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Two limit theorems for critical Bellman-Harris branching processes. (English. Russian original) Zbl 0569.60084
Math. Notes 36, 546-550 (1984); translation from Mat. Zametki 36, No. 1, 109-116 (1984).
Let \(\mu_ t\) be the number of particles alive at time t in a critical Bellman-Harris process. Under conditions on regular variation of the tails of the offspring distribution and lifetime distribution it is shown that for \(c_ 1<...<c_ n\leq 1\) the conditional distribution of \((\mu_{c_ 1t},...,\mu_{c_ nt})\) given \(\mu_ t>0\) exists and may be identified with the corresponding finite-dimensional distribution in a certain Markov process. The case \(n=1\), \(c_ 1=c_ n=1\) was treated earlier by V. A. Vatutin, Teor. Veroyatn. Primen. 22, 150-155 (1977; Zbl 0391.60082).
Reviewer: S.Asmussen
MSC:
60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
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References:
[1] B. A. Sevast’yanov, Branching Processes [in Russian], Nauka, Moscow (1971).
[2] V. A. Vatutin, ?Discrete limit distributions of the number of particles in critical Bellman-Harris branching processes,? Teor. Veroyatn. Primen.,26, No. 1, 150-155 (1977). · Zbl 0391.60082
[3] A. L. Yakymiv, ?Multidimensional Tauberina theorems and their application to Bellman-Harris branching processes,? Mat. Sb.,115, No. 3, 463-477 (1981).
[4] A. L. Yakymiv, ?Approximation of slowly varying functions by infinitely differentiable ones,? in: Certain Problems of Mathematics and Mechanics [in Russian], Moscow State Univ. (1981).
[5] E. Seneta, ?Regularly varying functions,? Lect. Notes Math.,508, Springer-Verlag, Berlin-Heidelberg (1976). · Zbl 0324.26002
[6] . L. Yakymiv, ?Limit theorem for branching processes,? Candidate’s Dissertation, Moscow State Univ. (1981).
[7] A. L. Yakymiv, ?Karamata, Keldysh and Littlewood type multidimensional Tauberian theorems,? Dokl. Akad. Nauk SSSR,270, No. 3, 558-561 (1983). · Zbl 0597.40005
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