Distribution of the branching-process population among generations
Distribution of the branching-process population among generations
复制标题
分支过程群体在各代之间的分布
DOI：
--
复制
发表时间：
1971
影响因子：
1
通讯作者：
M. Samuels
中科院分区：
文献类型：
--
作者：
M. Samuels
在标准的年龄相关分支过程中，设Rn (t)表示在时间t时属于第n代的总体比例。在超临界情况下，分布为{Rn (t)；对于大的t， n = 0,1，…}具有渐近的（非随机）正态形式，并且平均值ΣnRn (t)在t中渐近线性。进一步发现，对于大的n， Rn (t)具有(t)的正态密度函数的形状。还考虑了另外两个随机函数：(a)在时间t存活的第n代的比例，以及(b)在时间t出生的第n代的比例。这些函数也被发现具有渐近的范式，但其参数与{Rn (t)}的相关参数不同。对于临界和亚临界过程，用随机变量的期望值代替随机变量，也得到类似的结果。
Summary In a standard age-dependent branching process, let Rn (t) denote the proportion of the population belonging to the nth generation at time t. It is shown that in the supercritical case the distribution {Rn (t); n = 0, 1, …} has asymptotically, for large t, a (non-random) normal form, and that the mean ΣnRn (t) is asymptotically linear in t. Further, it is found that, for large n, Rn (t) has the shape of a normal density function (of t). Two other random functions are also considered: (a) the proportion of the nth generation which is alive at time t, and (b) the proportion of the nth generation which has been born by time t. These functions are also found to have asymptotically a normal form, but with parameters different from those relevant for {Rn (t)}. For the critical and subcritical processes, analogous results hold with the random variables replaced by their expectations.