### Asymptotics of the allele frequency spectrum associated with the Bolthausen-Sznitman coalescent.

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The stable fragmentation with index of self-similarity ∈[−1/2, 0) is derived by looking at the masses of the subtrees formed by discarding the parts of a (1+)−1–stable continuum random tree below height , for ≥0. We give a detailed limiting description of the distribution of such a fragmentation, ((), ≥0), as it approaches its time of extinction, . In particular, we show that 1/ ((−)+) converges in distribution as →0 to a non-trivial limit. In order to prove this, we go further and...

We extend Hoggar's theorem that the sum of two independent discrete-valued log-concave random variables is itself log-concave. We introduce conditions under which the result still holds for dependent variables. We argue that these conditions are natural by giving some applications. Firstly, we use our main theorem to give simple proofs of the log-concavity of the Stirling numbers of the second kind and of the Eulerian numbers. Secondly, we prove results concerning the log-concavity of the sum of...

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