Coalescence in the recent past in rapidly growing populations
Coalescence in the recent past in rapidly growing populations
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DOI：
10.1016/j.spa.2012.06.015
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发表时间：
2012-11-01
影响因子：
1.4
通讯作者：
Athreya, K. B.
中科院分区：
文献类型：
--
作者：
Athreya, K. B.
在一个快速增长的种群中，如果n很大，那么从第n代中随机选择的两个个体不太可能是密切相关的。在本文中，它表明，对于一类广泛的快速增长的人口，情况并非如此。对于后代分布{p（j）}满足p（0）= 0且psi（x）= Sigma（j）的Galton沃森分支过程，当x ->无穷大时，I-{j >= x}渐近于x（-alpha）L（x），其中L（.）在无穷远处缓慢变化，0 <α < 1（因此平均值m = Sigma jp（j）=无穷大），它表明，如果X-n是从第n代随机选择的两个个体的下降线在时间上向后合并的代数，则n-X-n在分布上收敛到由N = {1，2，3，...}支持的适当分布。也就是说，在如此快速增长的人口中，聚合发生在最近的过去，而不是遥远的过去。我们确实表明，如果后代平均值m满足1 < m，相当于Sigma jp（j）<无穷大，并且p（0）= 0，则聚结时间X-n确实收敛到n ->无穷大时的适当分布，即，聚结确实发生在遥远的过去(C)2012 Elsevier B. V.保留所有权利。
In a rapidly growing population one expects that two individuals chosen at random from the nth generation are unlikely to be closely related if n is large. In this paper it is shown that for a broad class of rapidly growing populations this is not the case. For a Galton Watson branching process with an offspring distribution {p(j)} such that p(0) = 0 and psi(x) = Sigma(j) p(j) I-{j >= x} is asymptotic to x(-alpha) L(x) as x -> infinity where L(.) is slowly varying at infinity and 0 < alpha < 1 (and hence the mean m = Sigma jp(j) = infinity) it is shown that if X-n is the generation number of the coalescence of the lines of descent backwards in time of two randomly chosen individuals from the nth generation then n - X-n converges in distribution to a proper distribution supported by N = {1, 2, 3, ...}. That is, in such a rapidly growing population coalescence occurs in the recent past rather than the remote past. We do show that if the offspring mean m satisfies 1 < m equivalent to Sigma jp(j) < infinity and p(0) = 0 then coalescence time X-n does converge to a proper distribution as n -> infinity, i.e., coalescence does take place in the remote past. (C) 2012 Elsevier B.V. All rights reserved.