?-measure for continuous state branching processes and its application |
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Authors: | Weijuan Chu Yan-Xia Ren |
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Institution: | 1. LMAM, School of Mathematical Sciences, Peking University, Beijing, 100871, China 2. Center for Statistical Science, Peking University, Beijing, 100871, China
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Abstract: | In this paper, we first give a direct construction of the ℕ-measure of a continuous state branching process. Then we prove,
with the help of this ℕ-measure, that any continuous state branching process with immigration can be constructed as the independent
sum of a continuous state branching process (without immigration), and two immigration parts (jump immigration and continuum
immigration). As an application of this construction of a continuous state branching process with immigration, we give a proof
of a necessary and sufficient condition, first stated without proof by M. A. Pinsky Bull. Amer. Math. Soc., 1972, 78: 242–244],
for a continuous state branching process with immigration to a proper almost sure limit. As another application of the ℕ-measure,
we give a “conceptual” proof of an L log L criterion for a continuous state branching process without immigration to have an L
1-limit first proved by D. R. Grey J. Appl. Prob., 1974, 11: 669–677]. |
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