Open Access
2018 Block size in Geometric($p$)-biased permutations
Irina Cristali, Vinit Ranjan, Jake Steinberg, Erin Beckman, Rick Durrett, Matthew Junge, James Nolen
Electron. Commun. Probab. 23: 1-10 (2018). DOI: 10.1214/18-ECP182


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.


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Irina Cristali. Vinit Ranjan. Jake Steinberg. Erin Beckman. Rick Durrett. Matthew Junge. James Nolen. "Block size in Geometric($p$)-biased permutations." Electron. Commun. Probab. 23 1 - 10, 2018.


Received: 14 August 2018; Accepted: 11 October 2018; Published: 2018
First available in Project Euclid: 24 October 2018

zbMATH: 1398.05004
MathSciNet: MR3873787
Digital Object Identifier: 10.1214/18-ECP182

Primary: 05A05 , 60J10 , 60K05

Keywords: Bernoulli sieve , regenerative permutations

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