Block size in Geometric(p)-biased permutations

TL;DR

This paper investigates the behavior of the first block in geometric and stretch exponential biased permutations, revealing asymptotic properties and conditions for finiteness of the block size.

Contribution

It provides the first analysis of block size distribution in geometric(p)-biased permutations and characterizes conditions for infinite blocks with stretch exponential distributions.

Findings

For geometric distribution, p log K converges to a constant as p approaches 0.

If p has a stretch exponential distribution, the block size K can be infinite with positive probability.

The results connect distribution tail behavior with block size properties.

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.