We study irreducible renewal matrices generated by matrices whose rows are proportional to various distribution functions. Such matrices arise in studies of multi-dimensional critical Bellman–Harris branching processes. Proofs of limit theorems for such branching processes are based on asymptotic properties of a chosen family of renewal matrices. In the theory of branching processes, unsolved problems are known that correspond to the case in which the tails of some of the above mentioned distribution functions are integrable, while the other distributions lack this property.We assume that the heaviest tails are regularly varying at the infinity with parameter −β ∈ [−1, 0) and asymptotically proportional, while the other tails are infinitesimal with respect to them. Under a series of additional conditions, we describe asymptotic properties of the first and second order increments for the renewal matrices.

中文翻译：

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与分支过程相关的具有不同阶分布尾部的更新矩阵

我们研究由行与各种分布函数成正比的矩阵生成的不可约更新矩阵。这种矩阵出现在多维临界Bellman–Harris分支过程的研究中。这种分支过程的极限定理的证明是基于所选更新矩阵族的渐近性质。在分支过程理论中，尚未解决的问题对应于上述某些分布函数的尾部是可积分的情况，而其他分布缺乏此性质的情况。我们假设最重的尾部在曲线上有规律地变化参数−β的无穷大∈[−1，0）且渐近成比例，而其他尾部相对于它们是无穷小的。在一系列其他条件下，我们描述了更新矩阵的一阶和二阶增量的渐近性质。