In this article, the asymptotic behavior of multitype Markov branching processes with discrete or continuous time is investigated in the positive regular and nonsingular case when both the initial number of ancestors and the time tend to infinity. Some limiting distributions are obtained as well as multivariate asymptotic normality is proved. The article also considers the relative frequencies of distinct types of individuals motivated by applications in the field of cell biology. We obtained non-random limits for the frequencies and multivariate asymptotic normality when the initial number of ancestors is large and the time of observation increases to infinity. In fact this paper continues the investigations of Yakovlev and Yanev [32] where the time was fixed. The new obtained limiting results are of special interest for cell kinetics studies where the relative frequencies but not the absolute cell counts are accessible to measurement.

中文翻译：

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多类型分支过程的限制分布

在本文中，研究了具有离散或连续时间的多类型马尔可夫分支过程在初始祖先数和时间都趋于无穷大的正正则和非奇异情况下的渐近行为。得到了一些极限分布，证明了多元渐近正态性。本文还考虑了受细胞生物学领域应用驱动的不同类型个体的相对频率。当祖先的初始数量很大并且观察时间增加到无穷大时，我们获得了频率和多元渐近正态性的非随机限制。事实上，本文继续了 Yakovlev 和 Yanev [32] 的研究，其中时间是固定的。