Block size in geometric(P)-biased permutations
© 2018, University of Washington. All rights reserved. Fix a probability distribution p = (p1, p2, …) on the positive integers. The first block in a p-biased permutation can be visualized in terms of raindrops that land at each positive integer j with probability pj. It is the first point K so that all sites in [1, K] are wet and all sites in (K, ∞) are dry. For the geometric distribution pj = 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 p has a stretch exponential distribution, then K is infinite with positive probability.
Cristali, I; Ranjan, V; Steinberg, J; Beckman, E; Durrett, R; Junge, M; Nolen, J
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