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  1. arXiv:1509.05199 (Published 2015-09-17)

    Singularity analysis for heavy-tailed random variables

    Nicholas M. Ercolani, Sabine Jansen, Daniel Ueltschi

    Let $\alpha \in (0,1),c>0$ and $G(z) = c\sum_{k=1}^\infty z^k \exp( - k^\alpha)$. We apply complex analysis to determine the asymptotic behavior of the $m$th coefficient of the $n$th power of $G(z)$ when $m,n\to \infty$ with $m \geq n G'(1)/G(1)$ and recover five theorems by A. V. Nagaev (1968) on sums of i.i.d. random variables with stretched exponential law $\mathbb{P}(X_1=k) = c \exp(- k ^\alpha)$. From the point of view of complex analysis, the main novelty is the combination of singularity analysis, Lindel\"of integrals, and bivariate saddle points. From the point of view of probability, the proof provides a new, unified approach to large and moderate deviations for heavy-tailed, integer-valued random variables.

  2. arXiv:1401.1442 (Published 2014-01-07, updated 2014-09-11)

    Random partitions in statistical mechanics

    Nicholas M. Ercolani, Sabine Jansen, Daniel Ueltschi
    Comments: 38 pages, 3 figures
    Journal: Electron. J. Probab. 19, 1-37 (2014)
    Categories: math.PR, math-ph, math.CO, math.MP
    Subjects: 60F05, 60K35, 82B05

    We consider a family of distributions on spatial random partitions that provide a coupling between different models of interest: the ideal Bose gas; the zero-range process; particle clustering; and spatial permutations. These distributions are invariant for a "chain of Chinese restaurants" stochastic process. We obtain results for the distribution of the size of the largest component.

  3. arXiv:1102.4796 (Published 2011-02-23, updated 2012-03-19)

    Cycle structure of random permutations with cycle weights

    Nicholas M. Ercolani, Daniel Ueltschi
    Comments: 22 pages, 2 figures
    Journal: Random Structures & Algorithms 44, 109-133 (2014)
    Categories: math.PR, math.CO
    Subjects: 60K35

    We investigate the typical cycle lengths, the total number of cycles, and the number of finite cycles in random permutations whose probability involves cycle weights. Typical cycle lengths and total number of cycles depend strongly on the parameters, while the distributions of finite cycles are usually independent Poisson random variables.