Abstract We consider Bienaymé-Galton-Watson and continuous-time Markov branching processes and prove diffusion approximation results in the near critical case, in fixed and random environment. In one hand, in the fixed environment case, we give new proofs and derive necessary and sufficient conditions for diffusion approximation to get hold of Feller-Jiřina and Jagers theorems. In the other hand, we propose a continuous-time Markov branching process with random environments and obtain diffusion approximation results. An averaging result is also presented. Proofs here are new, where weak convergence in the Skorohod space is proved via singular perturbation technique for convergence of generators and tightness of the distributions of the considered families of stochastic processes.

中文翻译：

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固定和随机环境中近临界分支过程的扩散近似

摘要 我们考虑 Bienaymé-Galton-Watson 和连续时间马尔可夫分支过程，并证明了在近临界情况下，固定和随机环境中的扩散近似结果。一方面，在固定环境的情况下，我们给出了新的证明并导出了扩散近似的充分必要条件，以得到 Feller-Jiřina 和 Jagers 定理。另一方面，我们提出了具有随机环境的连续时间马尔可夫分支过程并获得扩散近似结果。还提供了平均结果。这里的证明是新的，其中通过用于生成器收敛的奇异微扰技术和所考虑的随机过程族的分布的紧密性证明了 Skorohod 空间中的弱收敛。