Block size in Geometric(p)-biased permutations

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 $(K,\mathrm{\infty })$ are dry. For the geometric distribution $p_j= p(1-p)^{j-1}$weshowthat$p \log K$ 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.