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On the absolute continuity of the limit random variable in the supercritical Galton-Watson branching process


Author: Krishna B. Athreya
Journal: Proc. Amer. Math. Soc. 30 (1971), 563-565
MSC: Primary 60.67
DOI: https://doi.org/10.1090/S0002-9939-1971-0282421-5
MathSciNet review: 0282421
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Abstract: Let $ \{ {Z_n}:n \geqq 0\} $ be a simple Galton-Watson branching process with offspring distribution $ \{ {p_j}\} $ satisying $ 1 < \sum {j{p_j} < \infty } $. It is known that there exist constants $ {C_n}$ such that $ {W_n} \equiv {Z_n}{C_n}$ converges with probability one to a nondegenerate limit random variable W. Here we show that this W is always absolutely continuous on $ (0,\infty )$.


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DOI: https://doi.org/10.1090/S0002-9939-1971-0282421-5
Keywords: Galton-Watson branching process, supercritical, absolute continuity, limit random variable
Article copyright: © Copyright 1971 American Mathematical Society