Block size in Geometric(p)-biased permutations
Abstract
Fix a probability distribution $\mathbf p = (p_1, p_2, \cdots)$ 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.
- Publication:
-
arXiv e-prints
- Pub Date:
- August 2017
- DOI:
- 10.48550/arXiv.1708.05626
- arXiv:
- arXiv:1708.05626
- Bibcode:
- 2017arXiv170805626C
- Keywords:
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- Mathematics - Probability;
- 05A05;
- 60J10;
- 60K05
- E-Print:
- 10 pages, new title