Suppose k balls are dropped into n boxes independently with uniform probability, where n, k are large with ratio approximately equal to some positive real $\lambda$ . The maximum box count has a counterintuitive behavior: first of all, with high probability it takes at most two values $m_n$ or $m_n+1$ , where $m_n$ is roughly $\frac{\ln n}{\ln \ln n}$ . Moreover, it oscillates between these two values with an unusual periodicity. In order to prove this statement and various generalizations, it is first shown that for $X_1,\ldots,X_n$ independent and identically distributed discrete random variables with common distribution F, under mild conditions, the limiting distribution of their maximum oscillates in three possible families, depending on the tail of the distribution. The result stated at the beginning follows from the ensemble equivalence for the order statistics in various allocations problems, obtained via conditioning limit theory. Results about the number of ties for the maximum, as well as applications, are also provided.

中文翻译：

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一些离散过程的订单统计的振荡

认为ķ球落入n以均匀概率独立地框，其中n,ķ很大，比率大约等于某个正实数$\lambda$. 最大框数有一个违反直觉的行为：首先，它很有可能最多取两个值$m_n$要么$m_n+1$， 在哪里$m_n$大致是$\frac{\ln n}{\ln \ln n}$. 此外，它以不寻常的周期性在这两个值之间振荡。为了证明这个陈述和各种概括，首先证明对于$X_1,\ldots,X_n$具有共同分布的独立同分布离散随机变量F，在温和条件下，它们的最大值的极限分布在三个可能的族中振荡，这取决于分布的尾部。开头所述的结果来自于通过条件限制理论获得的各种分配问题中顺序统计的集合等价性。还提供了关于最大联系数以及应用程序的结果。