论文标题
连续数字的超级群集在$ p $降低的随机排列中
Super-clustering of consecutive numbers in $p$-shifted random permutations
论文作者
论文摘要
令$ a^{（n）} _ {l; k} \ subset s_n $表示$ l $ contecection number $ \ {k，k+1，\ cdots，k+l-1 \} $的事件出现在$ l $ l $ l $的位置中。令$ p = \ {p_j \} _ {j = 1}^\ infty $是$ \ mathbb {n} $上的分布，$ p_j> 0 $。令$ p_n $表示对应于$ p $降低随机排列的$ s_n $上的概率度量。我们的主要结果是在$ \ {p_j \} _ {j = 1}^\ infty $的额外假设之下，是$$ \ begin {Aligned}＆\ lim_ {lim_ {l \ to \ infty} \ to \ infty} \ lim_ { ）} _ {l，k}）= \ big（\ prod_ {j = 1}^{k-1} \ sum_ {i = 1}^jp_i \ big）\ big（\ prod_ {j = 1} $ \ lim_ {n \ to \ infty} \ min（k_n，n-k_n）= \ infty $，然后$$ \ begin {aligned}＆\ lim_ {l \ to \ infty} \ lim_ \ big（\ prod_ {j = 1}^\ infty \ sum_ {i = 1}^jp_i \ big）^2。 \ end {对齐} $$，特别是当且仅当$ \ sum_ {j = 1}^\ infty jp_j <\ infty $时，这些限制是正的。我们说，当限制为正时，就会发生超集群。我们还对$ s_ \ infty $上的$ p $降低概率分布的类别进行了新的特征。
Let $A^{(n)}_{l;k}\subset S_n$ denote the event that the set of $l$ consecutive numbers $\{k,k+1,\cdots, k+l-1\}$ appear in a set of $l$ consecutive positions. Let $p=\{p_j\}_{j=1}^\infty$ be a distribution on $\mathbb{N}$ with $p_j>0$. Let $P_n$ denote the probability measure on $S_n$ corresponding to the $p$-shifted random permutation. Our main result, under the additional assumption that $\{p_j\}_{j=1}^\infty$ is non-increasing, is that $$ \begin{aligned} &\lim_{l\to\infty}\lim_{n\to\infty}P_n(A^{(n )}_{l,k})=\big(\prod_{j=1}^{k-1}\sum_{i=1}^jp_i\big) \big(\prod_{j=1}^\infty\sum_{i=1}^jp_i\big), \end{aligned} $$ and that if $\lim_{n\to\infty}\min(k_n,n-k_n)=\infty$, then $$ \begin{aligned} &\lim_{l\to\infty}\lim_{n\to\infty}P_n(A^{(n )}_{l,k_n})= \big(\prod_{j=1}^\infty\sum_{i=1}^jp_i\big)^2. \end{aligned} $$ In particular these limits are positive if and only if $\sum_{j=1}^\infty jp_j<\infty$. We say that super-clustering occurs when the limits are positive. We also give a new characterization of the class of $p$-shifted probability distributions on $S_\infty$.
