The Martin boundary for Polya's urn scheme, and an application to stochastic population growth
The Martin boundary for Polya's urn scheme, and an application to stochastic population growth
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波利亚瓮方案的马丁边界及其在随机人口增长中的应用
DOI：
10.2307/3211860
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发表时间：
1964
影响因子：
1
通讯作者：
D. Kendall
中科院分区：
文献类型：
--
作者：
D. Blackwell;D. Kendall
1. 1923年，Eggenberger和Pólya提出了下面的“瓮计划”，作为传染现象发展的模型。一个盒子里有B个黑色球和r个红色球，从盒子里随机抽出一个球，进行“双替换”（即无论抽出什么球，它都会和一个相同颜色的新球一起回到盒子里）;然后这个过程无限地继续下去。有时讨论一个稍微复杂一些的m重置换方案，但对我们的目的来说，保持m = 2就足够了，而且取B = r = 1作为初始条件，进一步简化方案会很方便。然而，我们将在另一个方向上推广该方案，允许任意数量的颜色k（λ 2）。因此，最初的盒子将包含k个不同颜色的球，连续的随机抽取将像以前一样进行双重替换。我们写Sn（一个k向量，有第j个分量）来表示第n次替换后的盒子的数字组成，因此我们观察到这是一个马尔可夫过程，其状态空间由所有正整数的有序k-ad组成，（常数）转移概率矩阵的元素由下式确定其中Sn是Sn和（e（i））j = δ ij的分量之和。我们将计算这个马尔可夫过程的马丁边界，并指出一些人口增长的随机模型的应用。
1. In 1923 Eggenberger and Pólya introduced the following ‘urn scheme’ as a model for the development of a contagious phenomenon. A box contains b black and r red balls, and a ball is drawn from it at random with ‘double replacement’ (i.e. whatever ball is drawn, it is returned to the box together with a fresh ball of the same colour); the procedure is then continued indefinitely. A slightly more complicated version with m-fold replacement is sometimes discussed, but it will be sufficient for our purposes to keep m = 2 and it will be convenient further to simplify the scheme by taking b = r = 1 as the initial condition. We shall however generalise the scheme in another direction by allowing an arbitrary number k(≧2) of colours. Thus initially the box will contain k differently coloured balls and successive random drawings will be followed by double replacement as before. We write s n (a k-vector with jth component ) for the numerical composition of the box immediately after the nth replacement, so that and we observe that is a Markov process for which the state-space consists of all ordered k-ads of positive integers, the (constant) transition-probability matrix having elements determined by where S n is the sum of the components of s n and (e(i)) j = δ ij . We shall calculate the Martin boundary for this Markov process, and point out some applications to stochastic models for population growth.