Stochastic Comparisons Among Conditional Processes Derived from a Semiexplosive Galton-Watson Branching Process

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Let X= {Xn | n >) be a semi-explosive Glaton-Watson branching process, where ultimate extinction (Xn->0) and ultimate explosion (xn-> infinity) both occur with positive probability. Stochastic orderings among X and four conditional processes derived from it are studied: X = X |extinctiton, X = X | explosion, where X=X | no individual ever dies without offspring. It might be expected that but only four of these seven stochastic orderings hold in general. More refined results are given for special cases, including geometric and Poisson offspring distributions. An application to the problem of predicting extinction is noted.


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