论文标题
比较分布偏置和分布偏移排列的反演统计量与几何分布和GEM分布
Comparing the inversion statistic for distribution-biased and distribution-shifted permutations with the geometric and the GEM distributions
论文作者
论文摘要
对于分配$ p：= \ {p_k \} _ {k = 1}^\ infty $在正整数上，有两种自然的方法可以从$ s_n $或$ \ mathbb {n} $中构建来自$ p $ p $ $ p $ p $ by-bias-bias-bias-bias-bias-bias-bias-bias-bias-bias-bias-by-by-by-by-p $ p $ p $ p $ p $ p $ p $ p $ p $ p $ p $ p $ p $ p $ sshift $ p $ p $ p $ sships of $ s_nbb {n} $的自然置换。首先，我们认为$ p $是（0,1）$的参数$ 1-q \的几何分布。在这种情况下，$ p $缩短的随机排列具有带有参数$ q $的槌摩斯分布。令$ p_n^{b; \ text {geo}（1-q）} $和$ p_n^{s; \ text {geo}（1-q）} $表示有偏见的和$ s_n $的偏差分布。 $ p_n^{s; \ text {geo}（1-q）} $下置换的预期反转数大于$ p_n^{b; \ text {geo}（1-q）} $，并且在其中任何一个下，置换的逆转往往比均匀的分布较少。对于固定的$ n $，$ p_n^{b; \ text {geo}（1-q）} $和$ p_n^{s; \ text {geo}（1-q）} $在$ s_n $上的均匀分布中薄弱地将其弱收敛为$ q \ to。 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分销$ P $本身是随机的，并作为GEM $（θ）$ - 分布分布。在GEM $（θ）$ - 有偏见和GEM $（θ）$ - 移位的情况下，预期的反转数量渐近地行为，就像在GEO $（1-Q）$（1-Q）$转移的分布中一样，具有$θ= \ frac Q {1-Q} $。因此，人们可以将GEM $（θ）$ - 移位案例视为Geo $（Q）$转移案例的随机对应物。我们还考虑了另一个带有随机$ p $的$ p $偏分布，预期的反演数量在geo $（1-q）$ - 有$θ$和上述$ q $的情况下均匀地行为，并且具有上述$θ$，并且具有$θ\ \ fo \ to \ infty $和$ q \ q \ to1 $。
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 $q\to1$.
