几何分布和GEM分布下分布偏置和分布移位置换的反演统计量的比较
Comparing the inversion statistic for distribution-biased and distribution-shifted permutations with the geometric and the GEM distributions
摘要：
对于正整数上的分布$p:=\{p_k\}_{k=1}^\infty$，有两种自然的方法来构造$S_n$中的随机置换或来自$p$的IID样本的$\mathbb{N}$的随机置换--$p$偏置构造和$p$移位构造.首先我们考虑$p$是具有参数$1-q\in(0,1)$的几何分布的情况。在这种情况下，$p$移位的随机置换具有带有参数$q$的Mallow分布。令$P_n^{b;\text{Geo}(1-q)}$和$P_n^{s;\text{Geo}(1-q)}$表示$S_n$上的有偏分布和移位分布。在$P_n^{s;\text{Geo}(1-q)}$下的置换的期望反转数大于在$P_n^{b;\text{Geo}(1-q)}$下的置换的期望反转数，并且在这两种情况下，置换倾向于具有比在均匀分布下少得多的反转数。对于固定的$n$，$P_n^{b;\text{Geo}(1-q)}$和$P_n^{s;\text{Geo}(1-q)}$都弱收敛为$q\to1$到$S_n$上的均匀分布。我们通过研究在$P_n^{b;\text{Geo}(q_n)}$和$P_n^{s;\text{Geo}(q_n)}$下$q_n$收敛到1的各种速率下的反演统计来比较有偏和移位分布。然后，我们考虑$p$ -偏置和$p$ -移位排列的情况下，分布$p$本身是随机的，并作为GEM $(θ)$ -分布分布。在GEM $(θ)$偏置和GEM $(θ)$移位的情况下，反转的预期数量表现为渐近的，因为它在具有$θ=\frac q{1-q}$的Geo $(1-q)$移位分布下表现为渐近的。因此，可以将GEM $(θ)$移位情况视为Geo $(q)$移位情况的随机对应物.我们还考虑了另一个$p$有偏分布，其中随机$p$的期望反转数表现为渐近的，就像它在上面的$θ$和$q$以及$θ\to\infty$和$q\to1$的Geo $(1-q)$有偏情况下一样.
英文摘要：
For a distribution $p:=\{p_k\}_{k=1}^\infty$ on the positive integers, there are two natural ways to construct a random permutation in $S_n$ or of $\mathbb{N}$ from IID samples from $p$--the $p$-biased construction and the $p$-shifted construction. First we consider the case that $p$ is the geometric distribution with parameter $1-q\in(0,1)$. In this case, the $p$-shifted random permutation has the Mallows distribution with parameter $q$. Let $P_n^{b;\text{Geo}(1-q)}$ and $P_n^{s;\text{Geo}(1-q)}$denote the biased and the shifted distributions on $S_n$. The expected number of inversions of a permutation under $P_n^{s;\text{Geo}(1-q)}$ is greater than under $P_n^{b;\text{Geo}(1-q)}$, and under either of these, a permutation tends to have many fewer inversions than it would have under the uniform distribution. For fixed $n$, both $P_n^{b;\text{Geo}(1-q)}$ and $P_n^{s;\text{Geo}(1-q)}$ converge weakly as $q\to1$ to the uniform distribution on $S_n$. We compare the biased and the shifted distributions by studying the inversion statistic under $P_n^{b;\text{Geo}(q_n)}$ and $P_n^{s;\text{Geo}(q_n)}$ for various rates of convergence of $q_n$ to 1. Then we consider $p$-biased and $p$-shifted permutations in the case that the distribution $p$ is itself random and distributed as a GEM$(θ)$-distribution. In both the GEM$(θ)$-biased and the GEM$(θ)$-shifted cases, the expected number of inversions behaves asymptotically as it does under the Geo$(1-q)$-shifted distribution with $θ=\frac q{1-q}$. Thus, one can consider the GEM$(θ)$-shifted case as the random counterpart of the Geo$(q)$-shifted case. We also consider another $p$-biased distribution with random $p$ for which the expected number of inversions behaves asymptotically as it does under the Geo$(1-q)$-biased case with $θ$ and $q$ as above, and with $θ\to\infty$ and $q\to1$.
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