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Block size in geometric(\(p\))-biased permutations. (English) Zbl 1398.05004

Summary: Fix a probability distribution \(\mathbf{p} = (p_1, p_2, \ldots)\) on the positive integers. The first block in a \(\mathfrak p\)-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 \(\mathfrak p\) has a stretch exponential distribution, then \(K\) is infinite with positive probability.

MSC:

05A05 Permutations, words, matrices
60J10 Markov chains (discrete-time Markov processes on discrete state spaces)
60K05 Renewal theory
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References:

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