Consider an urn containing balls labeled with integer values. Define a discrete-time random process by drawing two balls, one at a time and with replacement, and noting the labels. Add a new ball labeled with the sum of the two drawn labels. This model was introduced by Siegmund and Yakir (2005) Ann. Prob.33, 2036 for labels taking values in a finite group, in which case the distribution defined by the urn converges to the uniform distribution on the group. For the urn of integers, the main result of this paper is an exponential limit law. The mean of the exponential is a random variable with distribution depending on the starting configuration. This is a novel urn model which combines multi-drawing and an infinite type of balls. The proof of convergence uses the contraction method for recursive distributional equations.

中文翻译：

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整数瓮中的随机加法

考虑一个装有标有整数值的球的瓮。定义一个离散时间随机过程，通过绘制两个球，一次一个并替换，并注意标签。添加一个标有两个绘制标签之和的新球。该模型由 Siegmund 和 Yakir (2005) 引入安。概率。33, 2036 对于在有限组中取值的标签，在这种情况下，由 urn 定义的分布收敛到组上的均匀分布。对于整数瓮，本文的主要成果是指数极限定律。指数的平均值是一个随机变量，其分布取决于起始配置。这是一种新颖的骨灰盒模型，它结合了多图和无限类型的球。收敛证明使用递归分布方程的收缩方法。