随机排列序的Erdens-Turán定律的推广
A generalization of the Erdős-Turán law for the order of random permutation
摘要：
我们考虑单位区间的破棒分割抽样得到的随机排列。这种置换的循环结构可以与$n$整数上的递减马尔可夫链的路径相关联。在一定的断棒因子假设下，我们证明了一个关于置换阶数对数的中心极限定理，从而推广了经典的Erdens-Turán一致置换定律及其在PD/GEM $(θ)$分布下Ewens置换的推广.我们的方法是基于使用扰动随机游动，以获得极限法律的总和的周期长度。
英文摘要：
We consider random permutations derived by sampling from stick-breaking partitions of the unit interval. The cycle structure of such a permutation can be associated with the path of a decreasing Markov chain on $n$ integers. Under certain assumptions on the stick-breaking factor we prove a central limit theorem for the logarithm of the order of the permutation, thus extending the classical Erdős-Turán law for the uniform permutations and its generalization for Ewens' permutations associated with sampling from the PD/GEM$(θ)$ distribution. Our approach is based on using perturbed random walks to obtain the limit laws for the sum of logarithms of the cycle lengths.
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