اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدExtinction probabilities in branching processes with countably many types: a general framework
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We consider Galton-Watson branching processes with countable typeset $mathcal{X}$. We study the vectors ${bf q}(A)=(q_x(A))_{xinmathcal{X}}$ recording the conditional probabilities of extinction in subsets of types $Asubseteq mathcal{X}$, given that the type of the initial individual is $x$. We first investigate the location of the vectors ${bf q}(A)$ in the set of fixed points of the progeny generating vector and prove that $q_x({x})$ is larger than or equal to the $x$th entry of any fixed point, whenever it is different from 1. Next, we present equivalent conditions for $q_x(A)< q_x (B)$ for any initial type $x$ and $A,Bsubseteq mathcal{X}$. Finally, we develop a general framework to characterise all emph{distinct} extinction probability vectors, and thereby to determine whether there are finitely many, countably many, or uncountably many distinct vectors. We illustrate our results with examples, and conclude with open questions.
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V.A.Topchii
, n V.A.Vatutin (Steklov Mathematical Institute of Russian Academy of Sciences
2020
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ymptotic behavior of the survival probability of the process and establish a Yaglom-type conditional limit theorem for the infinite-dimensional vector of the number of particles of all types.
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