Block size in Geometric($p$)-biased permutations

Electronic Communications in Probability

Irina Cristali¹ 

关键词: regenerative permutations;    Bernoulli sieve;   

DOI  :  10.1214/18-ECP182

学科分类：统计和概率

来源: Institute of Mathematical Statistics

【 摘 要 】

Fix a probability distribution $\mathbf p = (p_1, p_2, \ldots )$ on the positive integers. The first block in a $\mathbf p$-biased permutation can be visualized in terms of raindrops that land at each positive integer $j$ with probability $p_j$. It is the first point $K$ so that all sites in $[1,K]$ are wet and all sites in $(K,\infty )$ are dry. For the geometric distribution $p_j= p(1-p)^{j-1}$ we show that $p \log K$ converges in probability to an explicit constant as $p$ tends to 0. Additionally, we prove that if $\mathbf p$ has a stretch exponential distribution, then $K$ is infinite with positive probability.

【 授权许可】

CC BY